n 1 n 2 n 2 n n 1 ln n 2 1 n Express the number as a ratio of integers

Transcription

1 SECTION 8. SERIES 8. SERIES A Clck here for aswers. S Clck here for solutos. ; Fd at least 0 partal sums of the seres. Graph both the sequece of terms ad the sequece of partal sums o the same scree. Does t appear that the seres s coverget or dverget? If t s coverget, fd the sum. If t s dverget, expla why Determe whether the seres s coverget or dverget. If t s coverget, fd ts sum s e e s l.. 8 Express the umber as a rato of tegers Fd the values of x for whch the seres coverges. Fd the sum of the seres for those values of x. 9. x 0.. s x.. s s l ta x 0 x 0 x Copyrght 0, Cegage Learg. All rghts reserved.

2 SECTION 8. SERIES 8. ANSWERS E Clck here for exercses. S Clck here for solutos...,.,.88,.98,.99,.99,.99,.999,.999,.9999 Coverget, sum. 0.8,.08,.899,., 0.6, 0.0, 0.,.,.9,. Dverget terms do ot approach ,.666,.966,.66,.000,.0,.8, 6.0,.00,.980 Dverget terms do ot approach , , , 0., 0.600, , , 0., 0.000, 0.69 Coverget, sum , 0.86, 0.998, 0.9, 0.99, 0., 0.899, 0., 0.88, Dverget Dverget. 8. Dverget. e. Dverget π π. e e 9. Dverget 0. Dverget. 6. Dverget.. Dverget 6. Dverget. 8. Dverget s. Dverget.. l , <x< ; x x 0. <x<; x. π π 6 <x<π π 6 ay teger; s x x. x > ; x. π π <x<π π ay teger; ta x Coverget, sum 9 Copyrght 0, Cegage Learg. All rghts reserved.

3 SECTION 8. SERIES 8. SOLUTIONS E Clck here for exercses. Copyrght 0, Cegage Learg. All rghts reserved.... s s s From the graph, t seems that the seres coverges. I fact, t s a geometrc seres wth a 0 ad r,sotssum 0 s 0/. Note that the / dot correspodg to s part of both {a } ad {s }. The seres dverges, sce ts terms do ot approach 0. The seres dverges,sce ts terms do ot approach 0.. s From the graph, t seems that the seres 0. coverges to about. To fd the sum, we proceed as Example 6: sce , the partal sums are s adsothesums lm s.. s From the graph, t seems that the seres coverges to about 0.8. Ifact,tsa geometrc seres wth a ad r, so ts sum s / s a geometrc seres wth a ad r.sce r <, the seres coverges to / / s a geometrc seres wth a ad 8 r. Sce r >, the seres dverges.

4 SECTION 8. SERIES Copyrght 0, Cegage Learg. All rghts reserved. 8. a, r < so the seres coverges wth sum /. 9. s geometrc wth a, r,sot / coverges to /. 0. a 8, r 0 >, so the seres dverges a, r <, so the seres coverges wth sum 6 / 6 / dverges sce r 6 6 >.. a r <, sotheseres e e coverges to /e /e e.. For 8 the seres dverges , a 6 ad r 8 >,so a, r <, so the seres coverges to / a, r <, so the seres coverges to π /π π π.. a e, r e <, so the seres coverges to e/ e/ e e. 8. a, r 8 <, so the seres coverges to / so the seres dverges. 9, r >, 0. Ths seres dverges, sce f t coverged, so would by Theorem 8, whch we kow dverges Example.. Coverges. s / / partal fractos The latter sum s a telescopg seres: Thus, lm. [ These are coverget geometrc seres ad so by Theorem 8, ther 0. sum s also coverget lm a lm lm / 0, so the seres dverges by the Test for Dvergece... lm a lm by the Test for Dvergece. 6. lm / lm / 0, so the seres dverges // 0 so the seres dverges by the Test for Dvergece.. Coverges. s [ / / [ [ partal fractos [ [ 0 telescopg seres lm s 8. lm a lm does ot exst, so the seres dverges by the Test for Dvergece.

5 SECTION 8. SERIES Copyrght 0, Cegage Learg. All rghts reserved. 9. s [ / / partal fractos so lm s. 0. Coverges. s s s s s s s s s,so s s lm s s s 0 s. s l l l l l l [l l l l l telescopg seres. Thus, lm s, so the seres s dverget.. s / / / / / / both of whch are clearly telescopg sums, so [ [ s Thus, lm s.. Wrte l l.the s l l l l l l l Therefore, l lm s l l , x s geometrc wth r x, so coverges for 0 x < <x< to x. x x 0. s a geometrc seres wth r, so coverges wheever x < <x<. Thesums x/ x/ x x.. s x s geometrc so coverges wheever 0 sx < < s x< π π <x<π π,wherethesums 6 6 sx.. s geometrc wth r, so t coverges 0 x x wheever x < x > x>or x<, ad the sum s /x x x.. ta x s geometrc ad coverges whe ta x < 0 < ta x< π π <x<π π ay teger. O these tervals the sum s ta x.