8.6 Alternating Series

# 8.6 Alternating Series - 8 .6  A l t e r n a t in g  S...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 8 .6  A l t e r n a t in g  S e r ie s , A b s o l u t e  a n d  C o n d it io n a l   C o n v e rg e n c e   ∞ (−1) n +1 an   Alternating series have the form  ∑ n =1 a1 – a2 + a3 – a4 + ….    Alternating Series Test   ∞ (−1) n +1 an  converges if:  ∑ n =1 1. an > 0 for all n  2. an ≥ an+1  for all n ≥ N for some integer N  l 3.  nim an = 0   →∞   ∞ Ex.            ∑ (−1) n =1 n +1 1 n  ∞ Ex.  ∑ (−1) n =1 n +1 2n 2 4 n 2 + 1            Note:  The nth term test always shows divergence.  However, if  any of the other criteria fail in the alternating series test, it  does not mean that the series diverges.  (More at the end of  these overheads.)    Alternating Series Estimation Theorem  ∞ (−1) n +1 an  satisfies the three  If the alternating series  ∑ n =1 conditions of the Alternating Series Test, then for all n ≥ N  n sn = ∑ (−1) ai = a1 − a2 + a3 − ... + (−1) i =1 i +1 n +1 an   approximates the sum L of the series with an error whose  absolute value is less than an+1, the value of the first unused  term.   The remainder L – sn has the same sign as the first  unused term.  ∞ Ex.  ∑ (−1) n =1 n +1 n n 3 + 1              1.  Estimate the error in using the sum of the first four terms to  approximate the sum of the entire series.        2. Approximate the sum so the error has magnitude ≤ .01.                Do:  1. Are the following series convergent or divergent?  ∞ ∑ (−1) a.  n +1 n =1 ∞ ∑ c.  1 10 n   ∞ b.  1 n10   n =1 ∑ n =1 ∞ d.  1 10 n   ∑ (−1) n +1 n =1 1 10 − n     2.  How many terms of the series do we need to add in order to  find the sum if the error ≤.1?  ∞ ∑ (−1) n =1             n +1 1 2n − 1   Absolute and Conditional Convergence  ∞ A series  ∑ n =1 an  converges absolutely (is absolutely  convergent) if the corresponding series of absolute values,  ∞ ∞ ∑ an  , converges.   If  ∑ an n =1 ∞ ∑ an  converges then  n =1   converges.  (Sometimes a series is convergent, but the  alternating series test fails.  If you can show it is absolutely    convergent, then it is convergent.) n =1 ∞ Ex.  ∑ (−1) n +1 1 n2   n +1 1 3 n  n =1     ∞ Ex.  ∑ (−1) n =1       ∞ If  ∑ ∞ an ∑  diverges but  n =1 conditionally convergent. n =1 an  converges, then the series is  ...
View Full Document

## This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

Ask a homework question - tutors are online