# Homework Help: Sum to infinity question

1. Jan 22, 2014

### lionely

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

Sum to infinity

$$\frac{1}{2!} - \frac{ \pi ^2}{4^2.4!} + \frac{\pi^4}{4^4.6!} .....$$
2. Relevant equations

3. The attempt at a solution

I thought the series was similar to the Maclaurin expansion of cos x

so I tried putting in x= ∏/4

But I end up with the series $${1} - \frac{ \pi ^2}{4^2.2!} + \frac{\pi^4}{4^4.4!} .....$$

I don't know how to change the factorials to make it match the ones in the series given.

2. Jan 22, 2014

### haruspex

Try matching up the terms of the two series based on the factorials, not on the powers of pi. What do you notice?

3. Jan 22, 2014

### lionely

Well the factorials increase by 2. so it's like (2n)!?

4. Jan 22, 2014

### Integral

Staff Emeritus
You may want to look a bit closer at how you factored 1/2! out.

4! = 2! *3*4 <> 2*2!

5. Jan 22, 2014

### lionely

I know 4! is not equal to 2 x 2!. I didn't factor out anything all I did was put in x = pi/4 in the Maclaurin expansion of CosX

6. Jan 22, 2014

### pasmith

You can write this as $F(\pi/4)$, where
$$F(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+2)!}x^{2n}$$
so that
$$x^2F(x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+2)!}x^{2n+2} = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2(n+1))!}x^{2(n+1)}$$

7. Jan 22, 2014

### Staff: Mentor

Multiply the whole series by (π/4)2 and see what you get!

8. Jan 22, 2014

### haruspex

Yes. Compare the term with (2n)! in the first series with the term with (2n)! in the second series, same n.

9. Jan 22, 2014

### Staff: Mentor

$$s= \frac{1}{2!} - \frac{ \pi ^2}{4^2.4!} + \frac{\pi^4}{4^4.6!} .....$$

$$(\frac{π}{4})^2s= \frac{(\frac{π}{4})^2}{2!}-\frac{(\frac{π}{4})^4}{4!}+\frac{(\frac{π}{4})^6}{6!} ..... =1-\cos{\frac{π}{4}}$$

$$s=(\frac{4}{π})^2(1-\cos{\frac{π}{4}})=(\frac{4}{π})^2(1-\frac{1}{\sqrt{2}})$$

10. Jan 22, 2014

### lionely

Wow Chestermiller, I feel kind of stupid now. Thank you... I was thinking that it looked like cosx but the factorials weren't adding up, I should of tried to get the powers up so it could match. Thank you again guys.