{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 " " 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 286 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 287 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 267 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 268 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 269 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 270 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 271 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 272 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 273 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 274 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 275 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 276 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 277 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 278 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 279 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 280 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 281 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 282 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 283 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 284 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 285 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 286 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 287 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 288 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 289 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 290 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 291 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 292 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 293 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 294 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 295 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 296 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 297 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 298 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 299 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 300 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 301 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 302 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 303 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 304 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 305 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 306 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 307 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 308 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 309 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 310 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 311 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 312 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 313 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 314 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 315 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 316 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 317 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 318 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 319 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 320 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 321 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 322 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 324 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 326 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 328 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 330 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 331 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 332 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 333 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 334 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 335 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 336 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 337 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 346 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 347 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 349 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 350 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 351 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 352 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 353 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 354 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 355 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 356 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 357 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 358 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 359 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 360 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 361 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 362 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 363 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 364 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 365 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 366 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 367 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 368 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 369 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 370 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 371 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 372 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 373 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 374 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 375 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 376 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 377 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 378 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 379 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 380 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 381 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 382 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 383 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 384 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 385 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 386 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 387 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 388 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 389 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 390 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 391 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 392 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 393 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 394 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 395 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 396 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 397 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 398 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 399 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 400 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 401 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 402 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 403 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 404 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 405 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 3 406 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Armando H\344ring (11) Iso lde-Kurz-Gymnasium,Reutlingen" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 40 "Geometrische Folgen und Reihen mit Maple" }}}{SECT 1 {PARA 3 "" 0 "index" {TEXT 292 5 "Index" }}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "1.: Folgen" 1 "" "1.: Folgen" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Konvergenz u nd Divergenz" 1 "" "KON-g" }{HYPERLNK 17 "" 1 "" "KON-g" }}{PARA 0 "" 0 "" {HYPERLNK 17 "Eigenschaften einzelner Zahlenfolgen" 1 "" "Eigensc haften einzelner Zahlenfolgen" }}{PARA 0 "" 0 "" {HYPERLNK 17 "2.:Arit hmetische Folgen" 1 "" "Arithmetische Folge" }}{PARA 0 "" 0 "" {HYPERLNK 17 "3.:Geometrische Folgen" 1 "" "Geometrische Folge" }} {PARA 0 "" 0 "" {HYPERLNK 17 "4.:Grenzwerts\344tze" 1 "" "grenzwert" } {HYPERLNK 17 "" 1 "" "" }}}}{SECT 1 {PARA 269 "" 0 "" {TEXT -1 19 "1.: Folgen allgemein" }}{EXCHG {PARA 268 "" 0 "1.: Folgen" {TEXT 289 3 "1. :" }{TEXT 290 6 "Folgen" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 " restart: with (plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "Eine " } {TEXT 257 5 "Folge" }{TEXT -1 80 " ist eine geordnete Aneinanderreihun g von Zahlen.Die einzelnen Zahlen nennt man " }{TEXT 258 7 "Glieder" } {TEXT -1 12 " einer Folge" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 10 "Bei spiel: " }{TEXT -1 60 "Eine Folge bestehend aus den Gliedern 1, 4, 9, \+ 16, 25, .. ,." }}{PARA 0 "" 0 "" {TEXT -1 38 "Das erste Glied dieser F olge ist hier " }{XPPEDIT 18 0 "a[1]=1" "/&%\"aG6#\"\"\"\"\"\"" } {TEXT -1 14 " , das zweite " }{XPPEDIT 18 0 "a[2]=4 " "/&%\"aG6#\"\"# \"\"%" }{TEXT -1 147 " , usw. Jede Variable a wird mit einem Index (n) versehen und wird einem Glied der Folge zugeordnet. Das n-te Glied ei ner Folge bezeichnet man mit " }{XPPEDIT 18 0 "a[n]" "&%\"aG6#%\"nG" } {TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 144 "Die Zuordnung (Abst \344nde zwischen aufeinanderfolgende Glieder einer Folge) wird durch e ine Vorswchrift, meist durch eine Gleichung ausgedr\374ckt. " }} {PARA 0 "" 0 "" {TEXT -1 51 "Die Vorschrieft f\374r das oben erw\344hn te Beispiel ist:" }{XPPEDIT 18 0 "a[1]=1^2 ; a[2]=2^2 ; a[3]=3^2 ; usw . " "C%/&%\"aG6#\"\"\"*$\"\"\"\"\"#/&F%6#\"\"#*$\"\"#\"\"#/&F%6#\" \"$*$\"\"$\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 16 "Das n-te Glied " } {XPPEDIT 18 0 "a[n]" "&%\"aG6#%\"nG" }{TEXT -1 29 " dieser Folge ist \+ die Zahl " }{XPPEDIT 18 0 "n^2 " "*$%\"nG\"\"#" }{TEXT -1 30 " , als \+ Gleichung geschrieben: " }{XPPEDIT 18 0 "a[n]=n^2" "/&%\"aG6#%\"nG*$F& \"\"#" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "se q(a[i]=i^2,i=1..7); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6)/&%\"aG6#\"\" \"F'/&F%6#\"\"#\"\"%/&F%6#\"\"$\"\"*/&F%6#F,\"#;/&F%6#\"\"&\"#D/&F%6# \"\"'\"#O/&F%6#\"\"(\"#\\" }}}{EXCHG {PARA 258 "" 0 "KON-g" {TEXT -1 64 "Einschub zu dem Thema Konvergenz und Divergenz von Zahlenfolgen" }}{PARA 0 "" 0 "" {TEXT -1 33 "Zu beginn unter der \334berschrieft " } {TEXT 269 6 "Folgen" }{TEXT -1 71 " sprachen wir bei einer Zahlenfolge mit unbegrenzter Gliederanzahl als " }{TEXT 270 10 "unendliche" } {TEXT -1 1 " " }{TEXT 271 11 "Zahlenfolge" }{TEXT -1 71 ". Dabei k\366 nnen deren Glieder bei unbegrenzter Wachsender Gliedernummer " } {XPPEDIT 18 0 "n" "I\"nG6\"" }{TEXT -1 9 " entweder" }}{PARA 0 "" 0 " " {TEXT -1 37 "a) sich immer weiter einer eindeutig " }{TEXT 272 10 "a ngebbaren" }{TEXT -1 32 " Zahl (einem Limit) n\344hert, oder" }}{PARA 0 "" 0 "" {TEXT -1 11 "b) es sich " }{TEXT 273 18 "keine solche Zahl \+ " }{TEXT -1 112 "eindeutig angeben (z.B. wenn die Betr\344ge der Glied er mit wachsender Gliedernummer immer gr\366\337er werden)." }}{PARA 0 "" 0 "" {TEXT -1 63 " Unendliche Zahlenfolgen der unter a) b eschriebenen Art hei\337en " }{TEXT 274 10 "konvergent" }{TEXT -1 36 " , solche der unter b) erkl\344rten Art " }{TEXT 275 9 "divergent" } {TEXT -1 1 "." }}}{EXCHG {PARA 257 "" 0 "Eigenschaften einzelner Zahle nfolgen" {TEXT 260 47 "Besondere Eigenschaften einzelner Zahlenfolgen: " }}{PARA 0 "" 0 "" {TEXT -1 2 "1." }{XPPEDIT 18 0 "a[n+1]>a[n] " "2&% \"aG6#%\"nG&F$6#,&F&\"\"\"\"\"\"F*" }{TEXT -1 13 " f\374r alle n: " } {TEXT 261 33 "(streng monoton) wachsende Folge:" }{TEXT -1 1 " " } {XPPEDIT 18 0 "3,7,11" "6%\"\"$\"\"(\"#6" }{TEXT -1 10 " , .. ,415" }} {PARA 0 "" 0 "" {TEXT -1 2 "2." }{XPPEDIT 18 0 "a[n+1] " 0 " " {MPLTEXT 1 0 26 "seq(a[i+1]=a[i]+5,i=1..6);" }}{PARA 280 "" 1 "" {XPPMATH 20 "6(/&%\"aG6#\"\"#,&&F%6#\"\"\"F+\"\"&F+/&F%6#\"\"$,&F$F+F, F+/&F%6#\"\"%,&F.F+F,F+/&F%6#F,,&F3F+F,F+/&F%6#\"\"',&F8F+F,F+/&F%6#\" \"(,&F " 0 "" {MPLTEXT 1 0 14 "a[i+1]=a[i]+q;" }}{PARA 285 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#,& %\"iG\"\"\"F)F),&&F%6#F(F)%\"qGF)" }}}{EXCHG {PARA 286 "> " 0 "" {MPLTEXT 1 0 19 "Sum(a[i+1]=a[i]+q);" }}{PARA 287 "" 1 "" {XPPMATH 20 "6#-%$SumG6#/&%\"aG6#,&%\"iG\"\"\"F,F,,&&F(6#F+F,%\"qGF," }}}{EXCHG {PARA 288 "" 0 "" {TEXT -1 5 "Eine " }{TEXT 282 13 "arithmetische" } {TEXT -1 219 " Zahlenfolge n-ter Ordnung ist eine nicht konstante Zahl enfolge. Jedes Glied einer arithmetischen Folge erster Ordnung ist das arithmetische Mittel seiner beiden Nachbarglieder. Daraus erkl\344rt \+ sich auch die bezeichnung " }{TEXT 283 13 "arithmetische" }{TEXT -1 7 " Folge." }}}{EXCHG {PARA 289 "" 0 "" {TEXT 280 15 "Allgemein gilt:" } }{PARA 290 "" 0 "" {TEXT -1 90 "1. Bei arithmetischen Folgen 1.Ordnung entsteht eine lineare Funktion der Gliedernummer n." }}{PARA 291 "" 0 "" {TEXT -1 97 "2.Die konstante Zahlenfolge wird gew\366hnlich als \+ arithmetische Folge nullter Ordnung bezeichnet ." }}}{EXCHG {PARA 292 "" 0 "" {TEXT 281 19 "Beispiel mit Maple:" }}{PARA 293 "" 0 "" {TEXT -1 156 "Hier ein Beispiel mit Maple f\374r eine arithmetische Folge. D er konstante Wert q der jedesmal addiert wird ist hier 2. Ich beobacht e die Folge \374ber 6 Glieder." }}}{EXCHG {PARA 294 "> " 0 "" {MPLTEXT 1 0 21 "Zeit:=6:a[0]:=3:q:=2:" }}{PARA 295 "> " 0 "" {MPLTEXT 1 0 111 "for i from 0 to Zeit do a[i+1]:=a[i]+q od;\nfor i fr om 1 to Zeit do Punktea[i]:=plot([[i,a[i]]],style=point) od:" }}{PARA 296 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"\"\"\"&" }}{PARA 297 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"#\"\"(" }}{PARA 298 "" 1 "" {XPPMATH 20 "6 #>&%\"aG6#\"\"$\"\"*" }}{PARA 299 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\" %\"#6" }}{PARA 300 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"&\"#8" }}{PARA 301 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"'\"#:" }}{PARA 302 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"(\"#<" }}}{EXCHG {PARA 303 "" 0 "" {TEXT -1 96 "Der Graph wird aufgrund der errechneten Punkte die Form einer l inearen steigende Funktion haben." }}}{EXCHG {PARA 304 "> " 0 "" {MPLTEXT 1 0 63 "display(seq(Punktea[i],i=1..Zeit),title=`Arithmetisch e Folge`);" }}}{EXCHG {PARA 305 "" 0 "" {TEXT -1 116 "Jetzt verwende i ch die gleiche Folge nochmals, doch nun wird der Konstante Wert q nich t addiert sondern subtrahiert." }}}{EXCHG {PARA 306 "> " 0 "" {MPLTEXT 1 0 123 "for i from 0 to Zeit do a[i+1]:=a[i]-q od;\nfor i fr om 1 to Zeit do Punkteam[i]:=plot([[i,a[i]]],style=point,color=blue) o d:" }}{PARA 307 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"\"F'" }}{PARA 308 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"#!\"\"" }}{PARA 309 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"$!\"$" }}{PARA 310 "" 1 "" {XPPMATH 20 "6# >&%\"aG6#\"\"%!\"&" }}{PARA 311 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"&! \"(" }}{PARA 312 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"'!\"*" }}{PARA 313 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"(!#6" }}}{EXCHG {PARA 314 "" 0 "" {TEXT -1 52 "Diesmal ist das Ergebnis ein linear fallender Graph. " }}}{EXCHG {PARA 315 "> " 0 "" {MPLTEXT 1 0 80 "display(seq(Punkteam[ i],i=1..Zeit),title=`Arithmetische Folge mit negativem q`);" }}} {EXCHG {PARA 316 "" 0 "" {TEXT -1 64 "Hier ein Vergleich an dem man de n Unterschied gut erkennen kann." }}}{EXCHG {PARA 317 "> " 0 "" {MPLTEXT 1 0 85 "Vergleich[1]:=display(seq(Punkteam[i],i=1..Zeit)),dis play(seq(Punktea[i],i=1..Zeit)):" }}}{EXCHG {PARA 318 "" 0 "" {TEXT -1 66 "Hier f\374ge ich noch eine Information inden plot ein mit dem B efehl " }{HYPERLNK 17 "textplot" 2 "textplot" "" }}}{EXCHG {PARA 319 " > " 0 "" {MPLTEXT 1 0 234 "text:=[textplot([2,15,`rote Punkte = positi ves q`],align=\{ABOVE,RIGHT\})]:\ntext1:=[textplot([2,12,`blaue Punkte = negatives q`],align=\{ABOVE,RIGHT\})]:\ndisplay(Vergleich[1],title= `Vergleich der beiden arithmetischen Folgen`, text,text1)" }}}}{SECT 1 {PARA 271 "" 0 "" {TEXT -1 22 "3.:Geometrische Folgen" }}{EXCHG {PARA 260 "" 0 "Geometrische Folge" {TEXT -1 20 "Geometrische Folgen: " }}}{EXCHG {PARA 320 "> " 0 "" {MPLTEXT 1 0 22 "restart: with (plots) :" }}}{EXCHG {PARA 321 "" 0 "" {TEXT -1 45 "Die Zahl a hei\337t das An fangsglied der Folge (" }{XPPEDIT 18 0 "a[1]=a " "/&%\"aG6#\"\"\"F$" } {TEXT -1 61 "). Je zwei aufeinanderfolgende Glieder haben den Qutiente n q:" }}}{EXCHG {PARA 262 "" 0 "" {TEXT -1 9 "Beispiel:" }}}{EXCHG {PARA 322 "> " 0 "" {MPLTEXT 1 0 23 "seq(a[i+1]=i*4,i=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/&%\"aG6#\"\"#\"\"%/&F%6#\"\"$\"\")/&F%6#F( \"#7/&F%6#\"\"&\"#;" }}}{EXCHG {PARA 267 "" 0 "" {TEXT -1 30 "Darstell ung der Folge mit dem " }{HYPERLNK 17 "Sum" 2 "Sum" "" }{TEXT 287 8 "- Befehl:" }}}{EXCHG {PARA 324 "> " 0 "" {MPLTEXT 1 0 23 "Sum(a[i+1]=i*4 ,i=1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$/&%\"aG6#,&%\"iG \"\"\"F,F,,$F+\"\"%/F+;F,F." }}}{EXCHG {PARA 261 "" 0 "" {TEXT -1 15 " Allgemein gilt:" }}}{EXCHG {PARA 326 "> " 0 "" {MPLTEXT 1 0 14 "a[i+1] =a[i]*q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"aG6#,&%\"iG\"\"\"F)F) *&&F%6#F(F)%\"qGF)" }}}{EXCHG {PARA 328 "" 0 "" {TEXT -1 239 "Bei drei aufeinanderfolgenden Zahlen der Folge ist die mittlere Zahl das sog. \+ geometrische Mittel der beiden benachbarten Zahlen , falls alle Folgen glieder nicht negativ sind. Die Punkte einer geometrischen Folgen lieg en auf dem Graph der " }{TEXT 284 19 "Exponentialfunktion" }{TEXT -1 2 " :" }{MPLTEXT 1 0 13 "x->a*q^(x-1);" }{TEXT -1 7 ".Falls " } {XPPEDIT 18 0 "q>0" "2\"\"!%\"qG" }{TEXT -1 1 "." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#:6#%\"xG6\"6$%)operatorG%&arrowGF&*&%\"aG\"\"\")%\"qG,& 9$F,!\"\"F,F,F&F&" }}}{EXCHG {PARA 330 "" 0 "" {TEXT -1 95 "Man sprich t hier aber von einem geometrischen Wachstum statt von einem exponenti ellen Wachstum." }}{PARA 331 "" 0 "" {TEXT -1 132 "Anwendungen der geo metrischen Folgen und ihrer Summenfolge (gemetrische Reihe) findet man in Zinsrechnung und in der Rentenrechnung." }}}{EXCHG {PARA 332 "" 0 "" {TEXT 285 15 "Allgemein gilt:" }}{PARA 333 "" 0 "" {TEXT -1 96 "1. \+ Bei geometrischen Folgen 1.Ordnung entsteht eine exponentielle Funkti on der Gliedernummer n." }}{PARA 334 "" 0 "" {TEXT -1 96 "2.Die konsta nte Zahlenfolge wird gew\366hnlich als geometrische Folge nullter Ord nung bezeichnet ." }}}{EXCHG {PARA 335 "" 0 "" {TEXT 286 19 "Beispiel \+ mit Maple:" }}{PARA 336 "" 0 "" {TEXT -1 135 "Eine geometrische Folge \+ mit Maple. Der konstante Wert q mit dem multipliziert wird ist hier 2. Ich beobachte die Folge \374ber 12 Glieder." }}}{EXCHG {PARA 337 "> \+ " 0 "" {MPLTEXT 1 0 85 "restart:with(plots):\nZeit:=7:a[0]:=3:q:=2:\nf or i from 0 to Zeit do a[i+1]:=a[i]*q od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"\"\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>&%\"aG6#\"\"#\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"$ \"#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"%\"#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"&\"#'*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"'\"$#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >&%\"aG6#\"\"(\"$%Q" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\") \"$o(" }}}{EXCHG {PARA 266 "" 0 "" {HYPERLNK 17 "Zur Erinnerung Konver genz und Divergenz aus Kapitel 1 " 1 "" "KON-g" }}}{EXCHG {PARA 346 " " 0 "" {TEXT -1 95 "Anhand dieser if-Schleife kann ich vorab \374berpr \374fen ob die Folge konvergent oder divergent ist." }}}{EXCHG {PARA 347 "> " 0 "" {MPLTEXT 1 0 82 "if (q<=1) then `konvergent mit dem Gren zwert 0` fi;\nif (q>=1) then `divergent` fi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%*divergentG" }}}{EXCHG {PARA 349 "" 0 "" {TEXT -1 71 " Das Ergebnis ist hier ein Graph mit einer exponentiel ansteignden Form ." }}}{EXCHG {PARA 350 "> " 0 "" {MPLTEXT 1 0 121 "for i from 0 to Zei t do Pg[i]:=plot([[i,a[i]]],style=point) od:display(seq(Pg[i],i=0..Zei t),title=`Geometrische Folge`);\n" }}}{EXCHG {PARA 351 "" 0 "" {TEXT -1 25 "Wenn wir den Anfangswert " }{XPPEDIT 18 0 "a[0]" "&%\"aG6#\"\"! " }{TEXT -1 79 " negativ ist die Folge nat\374rlich exponentiel fallen d aber immer noch divergent." }}}{EXCHG {PARA 352 "> " 0 "" {MPLTEXT 1 0 156 "a[0]:=-3:\nfor i from 0 to Zeit do a[i+1]:=a[i]*q od:\nif (q< =1) then `Folge ist konvergent mit dem Grenzwert 0` fi;\nif (q>=1) the n ` Folge ist divergent` fi;" }}{PARA 353 "" 1 "" {XPPMATH 20 "6#%5~Fo lge~ist~divergentG" }}}{EXCHG {PARA 354 "> " 0 "" {MPLTEXT 1 0 147 "fo r i from 0 to Zeit do Pg[i]:=plot([[i,a[i]]],style=point) od:\ndisplay (seq(Pg[i],i=0..Zeit),title=`Geometrische Folge mit negativem Anfangsw ert`);" }}}{EXCHG {PARA 355 "> " 0 "" {MPLTEXT 1 0 8 "a[0]:=3:" }}} {EXCHG {PARA 265 "" 0 "" {TEXT 288 23 "Jetzt die Variante mit " } {XPPEDIT 18 0 " a[i]=a[0]*q^(i-1" "/&%\"aG6#%\"iG*&&F$6#\"\"!\"\"\")% \"qG,&F&F+\"\"\"!\"\"F+" }}}{EXCHG {PARA 356 "> " 0 "" {MPLTEXT 1 0 122 "for i from 0 to Zeit do a[i]:=a[0]*q^(i-1) od;\nfor i from 0 to Z eit do Pg1[i]:=plot([[i,a[i]]],style=point,color=blue) od:" }}{PARA 357 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"!#\"\"$\"\"#" }}{PARA 358 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"\"#\"\"$\"\"#" }}{PARA 359 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"#\"\"$" }}{PARA 360 "" 1 "" {XPPMATH 20 "6 #>&%\"aG6#\"\"$\"\"'" }}{PARA 361 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\" %\"#7" }}{PARA 362 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"&\"#C" }}{PARA 363 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"'\"#[" }}{PARA 364 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"(\"#'*" }}}{EXCHG {PARA 365 "> " 0 "" {MPLTEXT 1 0 103 "if (q<=1) then ` Folge ist konvergent mit dem Grenzw ert 0` fi;\nif (q>=1) then `Folge ist divergent` fi;" }}{PARA 366 "" 1 "" {XPPMATH 20 "6#%4Folge~ist~divergentG" }}}{EXCHG {PARA 367 "> " 0 "" {MPLTEXT 1 0 69 "display(seq(Pg1[i],i=0..Zeit),title=`Geometrisch en Folge mit q^i-1`);" }}}{EXCHG {PARA 368 "" 0 "" {TEXT -1 172 "Das E rgebnis ist wieder ein exponential Steigender Graph. Den Unterschied z u dem Vorhergehenden Graph erkennt erst deutlich wenn man beide Graphe n in einem plot vergleicht." }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 59 " Vergleich des Schaubildes mit dem vorhergehenden Schaubild:" }}} {EXCHG {PARA 369 "> " 0 "" {MPLTEXT 1 0 251 "Vergleich[2]:=display(seq (Pg[i],i=0..Zeit)),display(seq(Pg1[i],i=0..Zeit)):\ntext2:=textplot([2 ,12000,`rote Kreuze = Variante mit a[0]*q`],align=\{ABOVE,RIGHT\}):\nt ext3:=textplot([2,11000,`blaue Kreuze = Variante mit a[0]*q^(i-1)`],al ign=\{ABOVE,RIGHT\}):\n" }}}{EXCHG {PARA 370 "> " 0 "" {MPLTEXT 1 0 85 "display(Vergleich[2],title=`Vergleich der beiden Parabeln`,text2,t ext3,view=-50..50);" }}}{EXCHG {PARA 371 "" 0 "" {TEXT -1 77 "Deutlich zu erkennen das der vorhergehende Graph deutlich schneller ansteigt. " }}}{EXCHG {PARA 263 "" 0 "" {TEXT -1 26 "2.Variante mit negativem q " }}}{EXCHG {PARA 372 "" 0 "" {TEXT -1 84 "Jetzt machen wir das q mit \+ dem multiplieziert wird negativ, und schauen was passiert" }}}{EXCHG {PARA 373 "> " 0 "" {MPLTEXT 1 0 168 "Zeit:=7:a[0]:=3:q:=-2:\nfor i fr om 1 to Zeit do a[i+1]:=a[i]*q od;\nif (q<=1) then `Folge ist konverge nt mit dem Grenzwert 0` fi;\nif (q>=1) then `Folge ist divergent` fi; " }}{PARA 374 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"#!\"$" }}{PARA 375 " " 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"$\"\"'" }}{PARA 376 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"%!#7" }}{PARA 377 "" 1 "" {XPPMATH 20 "6#> &%\"aG6#\"\"&\"#C" }}{PARA 378 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"'!# [" }}{PARA 379 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"(\"#'*" }}{PARA 380 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\")!$#>" }}{PARA 381 "" 1 "" {XPPMATH 20 "6#%IFolge~ist~konvergent~mit~dem~Grenzwert~0G" }}}{EXCHG {PARA 382 "" 0 "" {TEXT -1 109 "Die Abfrage der if - Schleife ergab, d a\337 es sich jetzt um eine konvergente Folge mit dem Grenzwert 0 hand elt." }}}{EXCHG {PARA 383 "> " 0 "" {MPLTEXT 1 0 141 "for i from 1 to \+ Zeit do Pgm[i]:=plot([[i,a[i]]],style=point,color=black) od:\ndisplay( seq(Pgm[i],i=1..Zeit),title=`Variante mit negativem q`);" }}}{EXCHG {PARA 384 "" 0 "" {TEXT -1 202 "Das Ergebnis unterscheidet sich total \+ von den vorhergehenden Graphen. Diesmal ist der Graph nicht exponentia l steigend, sondern schl\344gt sowohl in y und -y Achse mit immer gr \366\337er werdenden Amplitude aus." }}}{EXCHG {PARA 385 "" 0 "" {TEXT -1 63 "Jetzt die Variante wie oben nur mit negativen q und der F ormel " }{XPPEDIT 18 0 "a[i]=a[0]*q^(i-1" "/&%\"aG6#%\"iG*&&F$6#\"\"! \"\"\")%\"qG,&F&F+\"\"\"!\"\"F+" }{TEXT -1 1 "." }}}{EXCHG {PARA 386 " > " 0 "" {MPLTEXT 1 0 206 "for i from 0 to Zeit do a[i]:=a[0]*q^(i-1) \+ od;\nfor i from 0 to Zeit do Pgm1[i]:=plot([[i,a[i]]],style=point,colo r=blue) od:\nif (q<=1) then `konvergent mit dem Grenzwert 0` fi;\nif ( q>=1) then `divergent` fi;" }}{PARA 387 "" 1 "" {XPPMATH 20 "6#>&%\"aG 6#\"\"!#!\"$\"\"#" }}{PARA 388 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"\"# !\"$\"\"#" }}{PARA 389 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"#\"\"$" }} {PARA 390 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"$!\"'" }}{PARA 391 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"%\"#7" }}{PARA 392 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"&!#C" }}{PARA 393 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\" '\"#[" }}{PARA 394 "" 1 "" {XPPMATH 20 "6#>&%\"aG6#\"\"(!#'*" }}{PARA 395 "" 1 "" {XPPMATH 20 "6#%?konvergent~mit~dem~Grenzwert~0G" }}} {EXCHG {PARA 396 "> " 0 "" {MPLTEXT 1 0 87 "display(seq(Pgm1[i],i=0..Z eit),title=`Variante mit negativen q und a[i]=a[0]*q^(i-1)`);" }}} {EXCHG {PARA 397 "" 0 "" {TEXT -1 119 "Gleiches Bild. Den unterschied \+ erkennt man erst wieder wirklich wenn man beide Graphen wieder in eine m plot vergleicht." }}}{EXCHG {PARA 273 "" 0 "" {TEXT -1 39 "Vergleich der Beiden neuen Schaubilder." }}}{EXCHG {PARA 398 "> " 0 "" {MPLTEXT 1 0 242 "Vergleich[3]:=display(seq(Pgm[i],i=1..Zeit)),display (seq(Pgm1[i],i=0..Zeit)):\ntext4:=textplot([2.5,3000,`schwarze Punkte \+ = Variante mit negativem q`]):\ntext5:=textplot([3,2500,`blaue Punkte \+ = Variante mit negativem q und a[i]=a[0]*q^(i-1)`]):" }}}{EXCHG {PARA 399 "> " 0 "" {MPLTEXT 1 0 75 "display(Vergleich[3],title=`Vergleich d er beiden Schaubilder`,text4,text5);" }}}{EXCHG {PARA 400 "" 0 "" {TEXT -1 140 "Der Vergleich zeigt, da\337 der Graph aus der ersten Var iante eine gr\366\337ere Amplitude als der aus Variante zwei. DerGraph aus Variante eins mit " }{XPPEDIT 18 0 "a[i]=a[0]*q" "/&%\"aG6#%\"iG* &&F$6#\"\"!\"\"\"%\"qGF+" }{TEXT -1 84 " hat jedesmal einen viel gr \366\337eren zuwachs pro Glied der Folge, als die Variante mit " } {XPPEDIT 18 0 "a[i]=a[0]*q^(i-1)" "/&%\"aG6#%\"iG*&&F$6#\"\"!\"\"\")% \"qG,&F&F+\"\"\"!\"\"F+" }{TEXT -1 5 " hat." }}}{EXCHG {PARA 274 "" 0 "" {TEXT -1 43 "Vergleich mit vorhergehendem Schaubild mit " } {XPPEDIT 18 0 "a[i]=a[0]*q" "/&%\"aG6#%\"iG*&&F$6#\"\"!\"\"\"%\"qGF+" }{TEXT -1 17 " mit positivem q." }}}{EXCHG {PARA 401 "> " 0 "" {MPLTEXT 1 0 77 "Vergleich[4]:=display(seq(Pgm1[i],i=0..Zeit)),display (seq(Pg[i],i=0..Zeit)):\n" }}}{EXCHG {PARA 402 "> " 0 "" {MPLTEXT 1 0 67 "text6:=textplot([2,12000,`rote Punkte = Parabel mit positivem q`]) :" }}}{EXCHG {PARA 403 "> " 0 "" {MPLTEXT 1 0 71 "text7:=textplot([3,1 0000,`schwarze Punkte = Parabel mit negativem q`]):" }}}{EXCHG {PARA 404 "> " 0 "" {MPLTEXT 1 0 85 "display(Vergleich[4],title=`Vergleich m it dem vorhergehendem Schaubild`,text6,text7);" }}}{EXCHG {PARA 405 " " 0 "" {TEXT -1 60 "Hier zeigt sich jetzt sch\366n welchen Unterschied es macht ob " }{XPPEDIT 18 0 "q" "I\"qG6\"" }{TEXT -1 27 " negativ o der positiv ist." }}}}{SECT 1 {PARA 406 "" 0 "" {TEXT 291 17 "4.:Grenz werts\344tze" }}{EXCHG {PARA 275 "" 0 "grenzwert" {TEXT -1 15 "Grenzwe rts\344tze:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:with (plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 191 "Bei der Bestimmung v on Grenzwerten gewisser Zahlenfolgen sind mitunter umformende Operatio nen erforderlich, die Beziehung zu den vier Grundrechenoperationen hab en. Sie werden durch vier sog. " }{TEXT 256 14 "Grenzwerts\344tze" } {TEXT -1 51 " festgelegt, die hier ohne Beweis mitgeteilt seien:" }} {PARA 0 "" 0 "" {TEXT -1 5 "Sind " }{XPPEDIT 18 0 "\{a[i]\}" "<#&%\"aG 6#%\"iG" }{TEXT -1 5 " und " }{XPPEDIT 18 0 "\{b[i]\}" "<#&%\"bG6#%\"i G" }{TEXT -1 52 " zwei konvergente Zahlenfolgen, d.h. existieren lim \+ " }{XPPEDIT 18 0 "a[i]" "&%\"aG6#%\"iG" }{TEXT -1 5 " und " }{XPPEDIT 18 0 "b[i]" "&%\"bG6#%\"iG" }{TEXT -1 66 " als eindeutigfeststellbare \+ Zahlen, so sind auch die Zahlenfolgen " }{XPPEDIT 18 0 "\{a[i]+b[i]" " <#,&&%\"aG6#%\"iG\"\"\"&%\"bG6#F'F(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " \{a[i]-b[i]" "<#,&&%\"aG6#%\"iG\"\"\"&%\"bG6#F'!\"\"" }{TEXT -1 2 ", \+ " }{XPPEDIT 18 0 " \{a[i]*b[i]" "<#*&&%\"aG6#%\"iG\"\"\"&%\"bG6#F'F(" }{TEXT -1 5 " und " }{XPPEDIT 18 0 "\{a[i]/b[i]" "<#*&&%\"aG6#%\"iG\" \"\"&%\"bG6#F'!\"\"" }{TEXT -1 94 " konvergent, und ihre Grenzwerte f \374r i = unendlich lassen sich aus den Grenzwerten der Folgen " } {XPPEDIT 18 0 "\{a[i]" "<#&%\"aG6#%\"iG" }{TEXT -1 5 " und " } {XPPEDIT 18 0 "\{b[i] " "<#&%\"bG6#%\"iG" }{TEXT -1 28 " folgenderma \337en berechnen : " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "F\374r die se Darstellung der vier Grenzwerts\344tze benutze ich den Befehl " } {HYPERLNK 17 "limit" 2 "limit" "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "limit(a[i]+b[i],i=infinity)=limit(a[i],i=infinity)+li mit(b[i],i=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&limitG6$, &&%\"aG6#%\"iG\"\"\"&%\"bGF*F,/F+%)infinityG,&-F%6$F(F/F,-F%6$F-F/F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "limit(a[i]-b[i],i=infinit y)=limit(a[i],i=infinity)-limit(b[i],i=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&limitG6$,&&%\"aG6#%\"iG\"\"\"&%\"bGF*!\"\"/F+%)infi nityG,&-F%6$F(F0F,-F%6$F-F0F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "limit(a[i]*b[i],i=infinity)=limit(a[i],i=infinity)*limit(b[i],i= infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&limitG6$*&&%\"aG6#% \"iG\"\"\"&%\"bGF*F,/F+%)infinityG*&-F%6$F(F/F,-F%6$F-F/F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "limit(a[i]/b[i],i=infinity)=limit(a [i],i=infinity)/limit(b[i],i=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%&limitG6$*&&%\"aG6#%\"iG\"\"\"&%\"bGF*!\"\"/F+%)infinityG*&-F %6$F(F0F,-F%6$F-F0F/" }}}{EXCHG {PARA 276 "" 0 "" {TEXT -1 46 "Grenzwe rtsatz f\374r eine divergente Zahlenfolge:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 23 "limit(a[i],i=infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limitG6$&%\"aG6#%\"iG/F)%)infinityG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "limit(a[i],i=-infinity);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&limitG6$&%\"aG6#%\"iG/F),$%)infinityG!\" \"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 180 "Mit diesen Grenzwerts\344t zen kann man in diesem Worksheet leider nicht rechnen, weil das umstel len des Worksheets zu diesem Zeitpunkt zu aufwendig gewesen w\344re. S ie sehen nur gut aus" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{EXCHG {PARA 272 "" 0 "" {HYPERLNK 17 "Zum Index" 1 "" "index" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Armando.Haering@ikg.rt.bw.schule.d e" }}}}{MARK "5" 0 }{VIEWOPTS 1 1 0 1 1 1803 }