Series III
<p><span style="font-size:18px;"><a href="https://www.prepswift.com/quizzes/quiz/prepswift-series-iii" target="_blank">Series III Exercise</a></span></p><h2>The summation operator</h2>
<p>A very common way to represent a series is by the summation operator, Σ. Here's how it works:</p>
<center><svg height="265" style="
width:662px;
height:265px;
background: #FFF;
fill: none;
" width="662" x="0" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" y="0"> <svg xmlns="http://www.w3.org/2000/svg"></svg> <svg class="role-diagram-draw-area" xmlns="http://www.w3.org/2000/svg"><g class="shapes-region" style="stroke: black; fill: none;"><g class="arrow-line"><path class="connection real" d=" M183,143 C222.6,113.3 242.6,171.81 281.81,143.87" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path><g stroke="#000" style="stroke: rgb(0, 0, 0); stroke-width: 1;" transform="matrix(-0.7999989281485084,0.6000014291326627,-0.6000014291326627,-0.7999989281485084,283.00000000000006,143)"><path d=" M10.93,-3.29 Q4.96,-0.45 0,0 Q4.96,0.45 10.93,3.29"></path></g></g><g class="arrow-line"><path class="connection real" d=" M307.56,131.92 L333,220" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path><g stroke="#000" style="stroke: rgb(0, 0, 0); stroke-width: 1;" transform="matrix(0.2774823366740286,0.960730738986695,-0.960730738986695,0.2774823366740286,307,130)"><path d=" M10.93,-3.29 Q4.96,-0.45 0,0 Q4.96,0.45 10.93,3.29"></path></g></g><g class="arrow-line"><path class="connection real" d=" M141,32 C180.8,2.15 207.73,2 264.15,72.93" stroke-dasharray="" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path><g stroke="#000" style="stroke: rgb(0, 0, 0); stroke-width: 1;" transform="matrix(-0.6207373542167665,-0.7840185821011952,0.7840185821011952,-0.6207373542167665,265,74)"><path d=" M10.93,-3.29 Q4.96,-0.45 0,0 Q4.96,0.45 10.93,3.29"></path></g></g><g></g></g><g></g><g></g><g></g></svg> <svg height="263" style="width:660px;height:263px;font-family:Asana-Math, Asana;background:#FFF;" width="660" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g><g><g><g><g><g><g transform="matrix(1,0,0,1,274.2125244140625,92.10001220703126)"><path d="M730 103L704 86C675 67 653 56 643 56C636 56 630 65 630 76C630 86 632 95 637 116L712 413C716 430 719 446 719 454C719 471 709 482 693 482C664 482 626 464 565 421C502 376 467 341 418 272L448 409C452 429 455 444 455 451C455 470 445 482 429 482C399 482 353 460 298 418C254 384 234 363 169 275L202 408C206 424 208 439 208 451C208 470 199 482 185 482C164 482 126 461 52 408L24 388L31 368C68 391 103 414 110 414C121 414 128 404 128 389C128 338 87 145 46 2L49 -9L117 6L139 108C163 219 190 278 245 338C287 384 332 414 360 414C367 414 371 406 371 393C371 358 349 244 308 69L292 2L297 -9L366 6L387 113C403 194 437 269 481 318C536 379 586 414 618 414C626 414 631 405 631 389C631 365 628 351 606 264C566 105 550 84 550 31C550 6 561 -9 580 -9C606 -9 642 12 740 85Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0119,0,0,-0.0119,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,268.7750244140625,134.07501831054688)"><path d="M24 388L31 368L63 389C100 412 103 414 110 414C121 414 128 404 128 389C128 338 87 145 46 2L53 -9C78 -2 101 4 123 8C142 134 163 199 209 268C263 352 338 414 383 414C394 414 400 405 400 390C400 372 397 351 389 319L337 107C328 70 324 47 324 31C324 6 335 -9 354 -9C380 -9 416 12 514 85L504 103L478 86C449 67 427 56 417 56C410 56 404 65 404 76C404 81 405 92 406 96L472 372C479 401 483 429 483 446C483 469 472 482 452 482C410 482 341 444 282 389C244 354 216 320 164 247L202 408C206 426 208 438 208 449C208 470 200 482 185 482C164 482 125 460 52 408ZM1159 347L1159 406L620 406L620 347ZM1159 134L1159 193L620 193L620 134ZM1463 722L1451 733C1399 707 1363 698 1291 691L1287 670L1335 670C1359 670 1369 663 1369 646C1369 638 1368 629 1367 622L1319 354C1305 279 1289 210 1237 1L1246 -9L1313 8L1336 132C1345 180 1369 225 1403 259C1472 59 1504 -9 1529 -9C1542 -9 1566 4 1610 35L1656 67L1648 86L1605 62C1591 54 1584 52 1576 52C1566 52 1559 58 1550 75C1515 139 1493 193 1454 316L1468 330C1529 391 1575 416 1625 416C1633 416 1644 414 1661 410L1678 473C1660 479 1642 482 1630 482C1564 482 1481 405 1356 230Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0119,0,0,-0.0119,0,0)"></path></g></g></g></g><g><g transform="matrix(1,0,0,1,268,113.17501831054688)"><path d="M788 13L772 -310L62 -310L420 133L66 620L754 620L768 346L732 346C718 428 694 486 662 522C626 559 572 578 498 578L246 578L524 194L214 -205L512 -205C592 -205 648 -190 680 -162C714 -132 734 -74 746 13Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0255,0,0,-0.0255,0,0)"></path></g></g></g><g><g transform="matrix(1,0,0,1,292.2249755859375,116.02499389648438)"><path d="M271 204L242 77C238 60 236 42 236 26C236 4 245 -9 260 -9C283 -9 324 17 406 85L399 106C375 86 346 59 324 59C315 59 309 68 309 82C309 87 309 90 310 93L402 472L392 481L359 463C318 478 301 482 274 482C246 482 226 477 199 464C137 433 104 403 79 354C35 265 4 145 4 67C4 23 19 -11 38 -11C75 -11 155 41 271 204ZM319 414C297 305 278 253 244 201C187 117 126 59 94 59C82 59 76 72 76 99C76 163 104 280 139 360C163 415 186 433 234 433C257 433 275 429 319 414Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,300.2625732421875,119.03751220703126)"><path d="M24 388L31 368L63 389C100 412 103 414 110 414C121 414 128 404 128 389C128 338 87 145 46 2L53 -9C78 -2 101 4 123 8C142 134 163 199 209 268C263 352 338 414 383 414C394 414 400 405 400 390C400 372 397 351 389 319L337 107C328 70 324 47 324 31C324 6 335 -9 354 -9C380 -9 416 12 514 85L504 103L478 86C449 67 427 56 417 56C410 56 404 65 404 76C404 81 405 92 406 96L472 372C479 401 483 429 483 446C483 469 472 482 452 482C410 482 341 444 282 389C244 354 216 320 164 247L202 408C206 426 208 438 208 449C208 470 200 482 185 482C164 482 125 460 52 408Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0119,0,0,-0.0119,0,0)"></path></g></g></g></g></g><g><g transform="matrix(1,0,0,1,312.987548828125,116.02499389648438)"><path d="M604 347L604 406L65 406L65 347ZM604 134L604 193L65 193L65 134Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,329.4500732421875,116.02499389648438)"><path d="M271 204L242 77C238 60 236 42 236 26C236 4 245 -9 260 -9C283 -9 324 17 406 85L399 106C375 86 346 59 324 59C315 59 309 68 309 82C309 87 309 90 310 93L402 472L392 481L359 463C318 478 301 482 274 482C246 482 226 477 199 464C137 433 104 403 79 354C35 265 4 145 4 67C4 23 19 -11 38 -11C75 -11 155 41 271 204ZM319 414C297 305 278 253 244 201C187 117 126 59 94 59C82 59 76 72 76 99C76 163 104 280 139 360C163 415 186 433 234 433C257 433 275 429 319 414Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,337.487548828125,119.03751220703126)"><path d="M240 722L228 733C176 707 140 698 68 691L64 670L112 670C136 670 146 663 146 646C146 638 145 629 144 622L96 354C82 279 66 210 14 1L23 -9L90 8L113 132C122 180 146 225 180 259C249 59 281 -9 306 -9C319 -9 343 4 387 35L433 67L425 86L382 62C368 54 361 52 353 52C343 52 336 58 327 75C292 139 270 193 231 316L245 330C306 391 352 416 402 416C410 416 421 414 438 410L455 473C437 479 419 482 407 482C341 482 258 405 133 230Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0119,0,0,-0.0119,0,0)"></path></g></g></g></g></g><g><g transform="matrix(1,0,0,1,347.4749755859375,116.02499389648438)"><path d="M604 241L604 300L364 300L364 538L305 538L305 300L65 300L65 241L305 241L305 0L364 0L364 241Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,362.237548828125,116.02499389648438)"><path d="M271 204L242 77C238 60 236 42 236 26C236 4 245 -9 260 -9C283 -9 324 17 406 85L399 106C375 86 346 59 324 59C315 59 309 68 309 82C309 87 309 90 310 93L402 472L392 481L359 463C318 478 301 482 274 482C246 482 226 477 199 464C137 433 104 403 79 354C35 265 4 145 4 67C4 23 19 -11 38 -11C75 -11 155 41 271 204ZM319 414C297 305 278 253 244 201C187 117 126 59 94 59C82 59 76 72 76 99C76 163 104 280 139 360C163 415 186 433 234 433C257 433 275 429 319 414Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,370.2750244140625,118.23749389648438)"><path d="M240 722L228 733C176 707 140 698 68 691L64 670L112 670C136 670 146 663 146 646C146 638 145 629 144 622L96 354C82 279 66 210 14 1L23 -9L90 8L113 132C122 180 146 225 180 259C249 59 281 -9 306 -9C319 -9 343 4 387 35L433 67L425 86L382 62C368 54 361 52 353 52C343 52 336 58 327 75C292 139 270 193 231 316L245 330C306 391 352 416 402 416C410 416 421 414 438 410L455 473C437 479 419 482 407 482C341 482 258 405 133 230ZM1072 241L1072 300L832 300L832 538L773 538L773 300L533 300L533 241L773 241L773 0L832 0L832 241ZM1554 -3L1554 27L1502 30C1447 33 1437 44 1437 96L1437 700L1196 598L1203 548L1353 614L1353 96C1353 44 1342 33 1288 30L1232 27L1232 -3C1386 0 1386 0 1397 0C1428 0 1538 -3 1554 -3Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0119,0,0,-0.0119,0,0)"></path></g></g></g></g></g><g><g transform="matrix(1,0,0,1,394.1375732421875,116.02499389648438)"><path d="M604 241L604 300L364 300L364 538L305 538L305 300L65 300L65 241L305 241L305 0L364 0L364 241Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,408.9000244140625,116.02499389648438)"><path d="M271 204L242 77C238 60 236 42 236 26C236 4 245 -9 260 -9C283 -9 324 17 406 85L399 106C375 86 346 59 324 59C315 59 309 68 309 82C309 87 309 90 310 93L402 472L392 481L359 463C318 478 301 482 274 482C246 482 226 477 199 464C137 433 104 403 79 354C35 265 4 145 4 67C4 23 19 -11 38 -11C75 -11 155 41 271 204ZM319 414C297 305 278 253 244 201C187 117 126 59 94 59C82 59 76 72 76 99C76 163 104 280 139 360C163 415 186 433 234 433C257 433 275 429 319 414Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,416.9375,118.23749389648438)"><path d="M240 722L228 733C176 707 140 698 68 691L64 670L112 670C136 670 146 663 146 646C146 638 145 629 144 622L96 354C82 279 66 210 14 1L23 -9L90 8L113 132C122 180 146 225 180 259C249 59 281 -9 306 -9C319 -9 343 4 387 35L433 67L425 86L382 62C368 54 361 52 353 52C343 52 336 58 327 75C292 139 270 193 231 316L245 330C306 391 352 416 402 416C410 416 421 414 438 410L455 473C437 479 419 482 407 482C341 482 258 405 133 230ZM1072 241L1072 300L832 300L832 538L773 538L773 300L533 300L533 241L773 241L773 0L832 0L832 241ZM1152 23L1152 -3C1339 -3 1339 0 1375 0C1411 0 1411 -3 1604 -3L1604 82C1489 77 1443 81 1258 77L1440 270C1537 373 1567 428 1567 503C1567 618 1489 689 1362 689C1290 689 1241 669 1192 619L1175 483L1204 483L1217 529C1233 587 1269 612 1336 612C1422 612 1477 558 1477 473C1477 398 1435 324 1322 204Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0119,0,0,-0.0119,0,0)"></path></g></g></g></g></g><g><g transform="matrix(1,0,0,1,440.800048828125,116.02499389648438)"><path d="M604 241L604 300L364 300L364 538L305 538L305 300L65 300L65 241L305 241L305 0L364 0L364 241Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,455.5625,116.02499389648438)"><path d="M124 111C94 111 67 83 67 53C67 23 94 -5 123 -5C155 -5 183 22 183 53C183 83 155 111 124 111ZM373 111C343 111 316 83 316 53C316 23 343 -5 372 -5C404 -5 432 22 432 53C432 83 404 111 373 111ZM622 111C592 111 565 83 565 53C565 23 592 -5 621 -5C653 -5 681 22 681 53C681 83 653 111 622 111Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,475.0625,116.02499389648438)"><path d="M604 241L604 300L364 300L364 538L305 538L305 300L65 300L65 241L305 241L305 0L364 0L364 241Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,489.8250732421875,116.02499389648438)"><path d="M271 204L242 77C238 60 236 42 236 26C236 4 245 -9 260 -9C283 -9 324 17 406 85L399 106C375 86 346 59 324 59C315 59 309 68 309 82C309 87 309 90 310 93L402 472L392 481L359 463C318 478 301 482 274 482C246 482 226 477 199 464C137 433 104 403 79 354C35 265 4 145 4 67C4 23 19 -11 38 -11C75 -11 155 41 271 204ZM319 414C297 305 278 253 244 201C187 117 126 59 94 59C82 59 76 72 76 99C76 163 104 280 139 360C163 415 186 433 234 433C257 433 275 429 319 414Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,497.862548828125,119.03751220703126)"><path d="M730 103L704 86C675 67 653 56 643 56C636 56 630 65 630 76C630 86 632 95 637 116L712 413C716 430 719 446 719 454C719 471 709 482 693 482C664 482 626 464 565 421C502 376 467 341 418 272L448 409C452 429 455 444 455 451C455 470 445 482 429 482C399 482 353 460 298 418C254 384 234 363 169 275L202 408C206 424 208 439 208 451C208 470 199 482 185 482C164 482 126 461 52 408L24 388L31 368C68 391 103 414 110 414C121 414 128 404 128 389C128 338 87 145 46 2L49 -9L117 6L139 108C163 219 190 278 245 338C287 384 332 414 360 414C367 414 371 406 371 393C371 358 349 244 308 69L292 2L297 -9L366 6L387 113C403 194 437 269 481 318C536 379 586 414 618 414C626 414 631 405 631 389C631 365 628 351 606 264C566 105 550 84 550 31C550 6 561 -9 580 -9C606 -9 642 12 740 85Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0119,0,0,-0.0119,0,0)"></path></g></g></g></g></g></g></g><g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:15px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="109" y="156.60000610351562">the starting index</text></g></g></g><g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:15px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="315" y="234.60000610351562">the term being added</text></g></g></g><g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:15px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="76" y="48.600006103515625">the last index to add</text></g></g></g></svg> </svg></center>
<p>Here is an example. Let $a_n = 2n$, so the sequence $2, 4, 6, ...$. Then, $$\sum_{i = 3}^{6} a_i = \sum_{i = 3}^{6} 2i = (6 + 8 + 10 + 12) = 36$$</p>
<blockquote>
<p>Keep in mind that you won't be expected to know the summation operator for the GRE and hence the above section can be considered as optional. ETS will instead describe the same thing without using the operator, so in the above example, one way they could phrase this would be "If the $n$<sup>th</sup> term of a sequence is $2n$, what is the sum of the third through the sixth term of the sequence?"</p>
</blockquote>
<h2>Telescoping series</h2>
<p>This is a common type of sequence where the terms cancel out in such a way that it becomes quite easy to find the series. The $n$<sup>th</sup> term of a telescoping sequence would look something like this:</p>
<p>$$a_n = \frac{1}{n} - \frac{1}{n + 2}$$</p>
<p>Why does this matter? Suppose we need to find the sum of the first $100$ terms of the sequence. Obviously it would be quite tedious to write down all $100$ terms, so (recalling from what we said in the previous entry) is there a pattern or shortcut we could use? Try writing down the first few terms:</p>
<p>$$\sum_{i = 1}^{100} \left(\frac{1}{i} - \frac{1}{i + 2}\right)$$</p>
<p>$$=1 - \frac{1}{3} +\frac{1}{2} - \frac{1}{4} + \frac{1}{3} - \frac{1}{5} + ...+ \left(\frac{1}{4} - \frac{1}{6}\right) ... $$</p>
<p>Look at the above equation carefully. Notice how most of the terms cancel out:</p>
<p>$$=1 \cancel{- \frac{1}{3}} +\frac{1}{2} \cancel{- \frac{1}{4}} + \cancel{\frac{1}{3}} \cancel{- \frac{1}{5}} + \cancel{\frac{1}{4}} \cancel{- \frac{1}{6}} ... $$</p>
<p>Which terms <em>don't</em> cancel out? $1$, $\frac{1}{2}$ and the last two terms to subtract (i.e, $\frac{1}{101}$ and $\frac{1}{102}$, because these two numbers don't have a corresponding positive part). As a result, the sum of the first $100$ terms would be </p>
<p>$$1 + \frac{1}{2} - \frac{1}{101} - \frac{1}{102}$$</p>
<p>$$\approx 1.4803$$</p>
<blockquote>
<p>Note that you won't always be given a telescoping series in that form (i.e, as a "difference of two fractions"), in which case you'll need to deduce that the series indeed is telescoping.</p>
</blockquote>
<hr />
<p>In the previous entry, we briefly touched upon the concept of an infinite series. The idea is that the as we sum up an increasingly large number of terms (towards infinity), the series will "approach" a certain value (i.e, <em>converges</em>). Let's extend the previous question to find the infinite series:</p>
<p>$$\sum_{i = 1}^{\infty} \left(\frac{1}{i} - \frac{1}{i + 2}\right)$$</p>
<p>The approach for doing this is pretty much the same as what we covered for finding the first $100$ terms, except that the "last two terms" to subtract wouldn't exist, because for very large $n$, $\frac{1}{n + 1}$ and $\frac{1}{n + 2}$ would both be pretty much $0$. So the "infinite sum" would be simply</p>
<p>$$1 + \frac{1}{2} = 1.5$$</p>
<p>One way to think of this is: if we sum up a finite number of terms, the series will approach and come very close to $1.5$, but won't reach (or go more than) $1.5$. </p>