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

1. May 4, 2012

### JJHK

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 an, Sn, and Rn, and find the limits of the sequences as n→∞.

2. Relevant equations

Sn is the partial sum of the series.

Rn is the remainder and is given by
Rn = S - Sn,
where
lim(n→∞) Sn = S

3. The attempt at a solution

an is pretty straightforward...

an = Ʃ(1→∞) (-1)n+1 (2n+1)n / [ (n) (n+1) ]

and as n→∞, an → 0

For Sn, I am getting:

Sn = 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)nx4

so as n→∞, does Sn approach 3 or -1, or neither?

Also, what would Rn be?

thank you so much!

2. May 4, 2012

### LCKurtz

Re: Convergence for a series: what is wrong with my method??

That is not $a_n$. What you have on the right is $S$, assuming the series converges.
$a_n$ is the nth term of the series$$a_n=\frac{-1^n(2n+1)^n}{n(n+1)}$$Unless you have mistyped the problem, I don't think you will find $a_n\rightarrow 0$. In any case you haven't shown any argument.
$S_n$ is the sum of the first $n$ terms. The expression for $S_n$ would never end with a "...". You can't calculate $S$ or $R_n=S-S_n$ unless the series converges. Go back to the beginning and make sure you copied the problem correctly and start by proving whether or not $a_n\rightarrow 0$.