8.2 Series

# 8.2 Series - R e v ie w  o f s e q u e n c e s  ...

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Unformatted text preview: R e v ie w  o f s e q u e n c e s   1 .  { a n }  is  a  s e q u e n c e  w h e r e  a n  is  t h e  n th t e r m  o f t h e   s e q u e n c e .  2 . A  s e q u e n c e  is  a  fu n c t io n .  l 3. A sequence converges if  nim an = L  where L is a finite  →∞ r e a l  n u m b e r .                              8 .2  I n fin it e  S e r ie s   An infinite series is an infinite sum.  ∞ ∑a n n =1 = a1 + a2 + a3 + ... + an + ....     an is the nth term of the series  n Sn = ∑ ai = a1 + a2 + a3 + ... + an i =1 partial sum.    ∞ Ex.              ∑n  n =1 ∞ 1 Ex.  ∑ n 3   n =1            is the nth  A geometric series has the form a + ar + ar2 + ar3 + … +arn + ….  ∞ or  ar n −1 ∑ n =1 ,  where a and r are real numbers, a ≠ 0.    Ex. 2 + 4 + 8 + 16 + 32 + ….                        ∞ 5 (−1) n   Ex.  ∑ 4 n =1                       n ∞ Def:  The infinite series  ∑a n =1 n  converges and has sum “s” if the  sequence of partial sums {Sn} converges to s.   If {Sn} diverges,  then the series diverges.    a if r < 1.  A geometric series converges to  1 − r The series diverges if |r| ≥ 1.                                            ∞ 2(2 ) Ex.  ∑   n −1 n =1         ∞ (−1) n Ex.  ∑ n =1 5 4n                   Do:  Write out the first few terms of the series.  Find a and r.   Then find the sum.  ∞ ∑                 n =1 n ⎛ 1⎞ 3 ⎝ 4⎠   Find the values of x for which the series converges.  Find the  sum of the series for those values of x.    ∞ ∑                           n =1 ∞ ∑                   n =0 (−1) n x n   ( x + 3) n 2n   Telescoping series    ∞ Ex.                        ∑ n =1 ∞ Ex.                          ∑ n =1 1 1 − n +1 n + 3  1 1 − ln( n + 2) ln( n + 1)   Tests for convergence:  l 1. Find an explicit formula for {sn}, then look at  nim →∞ ∞ Ex.  ∑n  n =1                   ∞ 1 Ex.  ∑ n   n =1             Sn     2. The nth Term Test for Divergence  ∞ ∑a n =1 n  diverges if  lim an  fails to exist or is different  n→∞ fr o m  z e r o .    ∞ ∑a Note:  If  n =1 n  converges, then  lim an = 0 .  However if  n→∞ lim an = 0 , the series does not necessarily converge.  n→∞ Y o u  m u s t  u s e  a n o t h e r  t e s t .    ∞ 1 1 lim = 0   but   diverges.  n→∞ n n =1 n ∑   ∞ n2 Ex.  ∑ 2 n 2 + 1   n =1         ∞ ⎛n⎞ ln Ex.  ∑ ⎝ 2 n + 1⎠   n =1       cos( nπ ) Ex.  ∑ 5n   n =0 ∞                                                         Rules:  ∞ 1.  2.  ∞ ∑ ka n =1 ∞ ∑a n =1 3.  If  = k ∑ an   n n n =1 ∞ ∞ n =1 n =1 ∞ ± bn = ∑ an ± ∑ bn   ∞ ∑a n =1 ∞ n  diverges then  4.   If  n  and  converges.  n =1 n  both converge, then  n  converges and  diverges.    ∞ If    n  and  ∑b n =1 ∞ Ex.      ∑n − n  n =1 ∞ Ex.  ∑ n + n  n =1 n =1 ∑b n =1 n n ± bn ∞ diverges, then  ∑a n =1 ∞ ∑a n =1 ∑a ∞ ∑a n =1 ∞ ∑b ∞ 5. If  n  diverges (k ≠ 0)  n =1 ∞ ∑a n =1 ∑ ka n  both diverge, it’s inconclusive.  n   ± bn     Do:  Determine if each of the following series converge or  diverge.  ∞ 1.  ∑ n =1 ∞ en   31 +n 2.  ∑ n 3  n =1 ∞ 3 3 − 3.  ∑ n n − 3  n =4 ...
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