1. ## Partial sum convergence

Hey all, I'm having a bit of trouble on this one:

Suppose $\sum^{\infty}_{1} a_{k}$ is a series with partial sums $S_{n} = \frac{1}{2n +1}, n \in N$

a) Explain why $\sum^{\infty}_{1} a_{k}$ is convergent and find the sum of the series.

b) Find $a_{k}$, k = 1, 2, ...

Some of the scratch work I've been doing:
$S_{1} = \frac{1}{3} = a_{1}$
$S_{2} = \frac{1}{5} = a_{1} + a_{2}$
$S_{3} = \frac{1}{7} = a_{1} + a_{2} + a_{3}$ ...

$a_{1} = \frac{1}{3}$
$a_{2} = -\frac{2}{15}$
$a_{3} = -\frac{2}{35}$

$\sum^{\infty}_{1} a_{k} = \frac{1}{3} - 2(\frac{1}{15} + \frac{1}{35} + \frac{1}{63} + ... + \frac{1}{(2n+1)(2n+3)} + ... )$

Something is obviously not jumping out at me. Any help would be appreciated.

2. The limit of the partial sums of a convergent series is the sum of the series.

3. Yeah I knew that much, but something in the back of my head was trying to tell me that that was way to easy.

Sn -> 0 right? Please tell me I'm wrong and there is actually some work to do here.

4. Unless it's a trick question, there doesn't seem to be much to say. I guess you could try and explain why a series converges iff its sequence of partial sums converges.

5. This is aggravatingly dumb. I'll ask the professor tomorrow I guess.

Alright then, so what is $a_{k}$, k = 1,2... ?

6. The solution in implicit in definition of 'partial sum' ...

$S_{n} = \sum_{k=1}^{n} a_{k}$ (1)

From (1) we derive immediately...

$a_{n} = S_{n} - S_{n-1} = \frac{1}{2n+1} - \frac{1}{2n-1} = \frac{2}{1-4n^{2}}$ (2)

The series of the $a_{n}$ is convergent and its sum is...

$S=\lim_{n \rightarrow \infty} S_{n} = 0$ (3)

Kind regards

$\chi$ $\sigma$