Homework Help: Find the sum of the following series

1. May 7, 2006

verd

Hi,

I'm having a bit of difficulty with the following two problems:

The first asks to find the sum of the following series:
$$\sum\limits_{n = 1}^\infty {\frac{2}{n^2+4n+3}}$$

I easily understand geometric series and how to find sums with those, but this seems a bit complicated. Factoring and pulling the 2 out doesn't seem to really help any. ...Any suggestions?

And this question:

Find a power series representation for the following:
$$g(x)=\arctan{\frac{x}{2}}$$

I understand the simple 1/4+x^2 kind of stuff, but this I don't even know how to begin.

Any suggestions??

Thanks!

2. May 7, 2006

verd

...Telescoping series for #1??

3. May 7, 2006

Geekster

I would suggest checking to see if partial fraction decomposition on the summand makes the problem easier to get at.

After you break it apart you should get something like

$$\sum\limits_{n = 1}^\infty {\frac{1}{n+1}} - \sum\limits_{n = 1}^\infty {\frac{1}{n+3}}$$

Then notice that this telescoping series has only two terms that don't get cancelled out. Add those two terms to get the sum.

Last edited: May 7, 2006
4. May 7, 2006

verd

Wouldn't that work for testing convergence/divergence?

I need to find the sum... I wasn't aware that by partial fractions/integrals you could find sums... Is that possible? If so, what are the conditions?

5. May 7, 2006

Geekster

What's the derivative of arctan(x)?

Can you find a power series representation for that? Can you integrate term by term?

6. May 7, 2006

Geekster

That works fine for the sum. The partial faction decomp isn't just for integrals. Try adding the two sums together and see that you get back your original summand. Once you're convinced they are in fact the same, then notice that every n+2 for the first summand is the additive inverse of the nth term of the second summand. You should notice that only two terms are not cancelled out in this manner. Add those two terms to get the sum.

7. May 7, 2006

verd

Woo, I figured it out right before you replied! Haha, thanks, you verified that what I was doing was correct! Thanks!

Anyone have any ideas on that second one?

8. May 7, 2006

Geekster

Look three post up from this one...

9. May 8, 2006

verd

Can someone tell me if this is correct for the second problem posted?

$$\sum\limits_{n = 1}^\infty {\frac{(-1)^{n}x^{2n-1}}{(2n-1)4^{n-1}}}$$

10. May 8, 2006

Curious3141

The easiest way (non-rigorous, but fast) to verify your answer is to work out around five to six terms with some "not so nice" value for x (like 0.23) and confirm that the sum is tending to the correct value for arctan(0.23/2). Quite fast with a scientific calc and the Memory+ function.