Infinite Series Study Resources

10.2 Infinite Series 53 * An infinite series is the sum of an infinite sequence of numbers a, + ar + az + ... + ant... Zan where an is the nth term of the series st is the 1 partial sum of the series • 2nd partial sum of the series 2 = a + az is the 2 Sn = a + ar ++ an = Σακ ak is the th partial sum of series. If the sequence and we write of partial sums converges to a limit I then we say the series convergs. a + az ++ an t٠٠ = an = L 00 Ean If the sequence sequence of partial sums of the serie does not Converge, then we say the series diverges. EXP Partial sums First 5₁ = 1 second sz=1+1=2- Thire 53 = 1++=2- nth Sn = 1+++++(+) = 2- •Note that this sequence of partial sums converges to 2 because lim Sn = 2-0 2 00 n-1 • Thus, we Say the sum of the in finite series () = 2 Geometric Series 54 Geometric series are series of the form: a + ar + ar² +...+ ar² Σα ar where •a and r are fixed real numbers • a ≠0 is called the ratio and 1+++……+(½)... = 4 or negative: Can be positive: a = 1 9=1 = To determine 1-3 + convergoure and divergence of the + geemeie series #; we cons * If ral, then the nth partial sum of the n-1 geometric series is and the series Sn = a + a (1) + a(1)²+...+(1) = na 2 diverges because lim Sn = ± 00 on the sign of a. If r=-1, then the nth partial i's sum of the series Sn = d -4 +d-x+a... a(-1)". ta io ifn even a odd Thus, the series diverges because the nth partial sum alternate between a and = if n • If r≠ 1 and r≠-1 The Convergence Sn = a + ar at + rSn Sn-rsna- ar Sn: a(1-r") ۱-۷ + (ie Ir171) then we can or divergence as follows: ar tar 7-1 + ar Sn (1-x)=a(1-r") Jelerimme • If Irl, then ro as n 1-4 55 Thus Ear converges to li'm Sn a = ۱-۷ and the series diverges • If Irl>', then→∞ * If /rk, the geometric series at ar to a 2 и-1 + ar ++ ar +... : ۸-۱ 2 ar converges a 1시 If Irla, the series diverges и-1 00 +++..++...: Ear" = {(½)" -1 Exp: Express the repating decimal numbers 23 as the ratio of two integers: 4-1 9 10 1=4 00 1-и = 2 ①0.23 0.23 23 23 ... = 100 + 23 + (100) (100) 3 )3= 100 0.7 = 0.777... 7 = + (10) + 7 -1 10 ③0.06 0.0666... = 6 6 6 100 1000 10000 The nth Term Test for Divergent Series 56 00 * If lim an i'm a fails to exist or is different from zero, then an diverges n-300 Ex The series no diverges because lim an = lim "t=1 The series Ein diverges becuse lim an = lim n = as The series (1) diverges because lim (-1)"'t does not exist n+1 The series Σ -n 27+5 diverges because lim=-≠0 100 21+6 Th? If an converges, n=1 then lim an = 0. Note that th7 does not say that if lim an o then an Converges exp 8W 00 harmonic" Series The If an A and bn=B are convergent series; then Sum Rule: Σ(ant br) = Ean+Ebn = A +B 2 Difference Rule Constant Multiple Rules Σ(an-bn) = Ean-Ebn = A-B Ek an= Ean K KA, KEIR Nole that Every nonzero cons 2 constant multiple of divergent series is Divergan If Ean converges and Ebn diverges, then Elantby) and Elan-bu) both diverges. "Telescoping Series" a formula for the nth partial sum of the following Exe Find series and use it to determine if the series convergos or diverges. If the series converges find the sum. ①吃(一) 8 ۱ Sn = 1 - = lim Su: 1. Thus, the sentes Vn+1 converges to 1 ie (一点)=1 1 n(n+1) " we use partial fraction" " 2 00 + A=1 n(n+1) - lim Sn = 1. Thus, the series converges to 1. 40 iencaru Σ1 = 1 n(n+1) Ene Find the sum of the following series. (+)=(5+1)+(+)+(+)+(+)+... 2 = 5 1-5 = + [루부+두+1]+[+++]= = 10 +올을 =