Series Converges to?

1. The problem statement, all variables and given/known data
I have the series [tex]\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)!}[/tex] 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.


[tex]\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)!}=\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)(2n+1)!}[/tex], from which you can differentiate term by term
Why would I differentiate it? How does that help find what it converges to?


[tex]\frac{d}{dx}\sum\frac{b^{(2n+2)}(-1)^{n}}{(2n+2)(2n+1)!}=\sum\frac{b^{(2n+1)}(-1)^{n}}{(2n+1)!}[/tex] which looks a bit like which function?
It looks like the general term of the Taylor Polynomail for sin(x) it is sin(b)?


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

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