(adsbygoogle = window.adsbygoogle || []).push({}); Convergence for a series: what is wrong with my method??

1. The problem statement, all variables and given/known data

For the following series, write formulas for the sequence a_{n}, S_{n}, and R_{n}, and find the limits of the sequences as n→∞.

2. Relevant equations

S_{n}is the partial sum of the series.

R_{n}is the remainder and is given by

R_{n}= S - S_{n},

where

lim(n→∞) S_{n}= S

3. The attempt at a solution

a_{n}is pretty straightforward...

a_{n}= Ʃ(1→∞) (-1)^{n+1}(2n+1)^{n}/ [ (n) (n+1) ]

and as n→∞, a_{n}→ 0

For S_{n}, I am getting:

S_{n}= 3 / (1x2) - 5 / (2x3) + 7 / (3x4) - 9 /(4x5) + ....

= 3 (1/1 - 1/2) - 5 (1/2 - 1/3) + 7(1/3 - 1/4) - 9 (1/4 - 1/5) + ...

= 3 - 1/2 (3+5) + 1/3 (5+7) - 1/4 (7+9) + ...

= 3 - 8/2 + 12/3 -16/4 + ....

= 3 - 4 + 4 - 4 + ....

= 3 + Ʃ(1→∞) (-1)^{n}x4

so as n→∞, does S_{n}approach 3 or -1, or neither?

Also, what would R_{n}be?

thank you so much!

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# Homework Help: Convergence for a series: what is wrong with my method?

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