# Series Converges to?

#### Swerting

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

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

3. 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.

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