Series Converges to b: Find Method

[Swerting]

Homework Statement

I have the series $$\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)!}$$ from n=0 to infinity. I am trying to find what it converges to in terms of b.

Homework Equations

Using the Ratio Test I have established that it does converge.

The Attempt at a Solution

I have scoured the internet, my notes, and all my books, but I can't seem to find a way to find what these kinds of series (power I believe) converge to, only ways to see if they converge or not. I just need to find out the method to calculate what it converges to in terms of b. Thank you for any assistance.

 

[zcd]

$$\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)!}=\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)(2n+1)!}$$, from which you can differentiate term by term

 

[Swerting]

Why would I differentiate it? How does that help find what it converges to?

 

[zcd]

$$\frac{d}{dx}\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)(2n+1)!}=\sum\frac{b^{(2n+1)}(-1)^{n}}{(2n+1)!}$$ which looks a bit like which function?

 

[Swerting]

It looks like the general term of the Taylor Polynomail for sin(x)...so it is sin(b)?

 

[zcd]

The derivative of the series converges to sin(b), not the original series.