General ways or ideas for finding a function from its power/taylor series I am a Calculus 2 student and came across multiple problems requiring me to find the elementary function from its Taylor series. However, the teaching materials, including my reference book, do not have anything related to finding a function from its power/taylor series. I have tried to look it up online and found that I could do differentiation or integration. However, this method didn't always work when I was struggling with my problem set. What are some general ways or stuff I can think about when I come across this type of problem?
Here is one of my questions:
Find the elementary function of the following series:
$$\sum_{n=0}^{\infty}\frac{(-1)^{n+1}(2n+2)x^{2n+1}}{(2n)!} $$
 A: In order to transform the series we need one additional technique besides differentiation/integration.

We obtain
\begin{align*}
\color{blue}{\sum_{n=0}^{\infty}}&\color{blue}{\frac{(-1)^{n+1}(2n+2)x^{2n+1}}{(2n)!}}\\
&=\frac{d}{dx}\left(\sum_{n=0}^{\infty}(-1)^{n+1}\frac{x^{2n+2}}{(2n)!}\right)\\
&=-\frac{d}{dx}\left(x^2\sum_{n=0}^{\infty}\frac{(ix)^{2n}}{(2n)!}\right)\\
&=-\frac{d}{dx}\left(x^2\sum_{n=0}^{\infty}\color{blue}{\frac{1+(-1)^n}{2}}\,\frac{(ix)^n}{n!}\right)\tag{*}\\
&=-\frac{d}{dx}\left(\frac{x^2}{2}\sum_{n=0}^{\infty}\frac{(ix)^n}{n}+\frac{x^2}{2}\sum_{n=0}^{\infty}\frac{(-ix)^n}{n!}\right)\\
&=-\frac{d}{dx}\left(x^2\,\frac{e^{ix}+e^{-ix}}{2}\right)\\
&=-\frac{d}{dx}\left(x^2\cos(x)\right)\\
&\,\,\color{blue}{=-2x\cos(x)+x^2\sin(x)}
\end{align*}

We observe besides knowing the series expansion of some standard functions like $e^x, \sin(x), \cos(x)$ we need as additional technique besides differentiation the so-called multisection (*) of series. See for instance this MSE post for additional info. Here we have the simplest form of multisection, namely subdivision in even and odd parts via
\begin{align*}
\frac{1+(-1)^n}{2}
\end{align*}
Hint:

*

*A collection of useful techniques can be found when going through chapter 2 of generatingfunctionology by H. S. Wilf.


*Section 2.5 Some useful power series contains a list of series expansions which is helpful to keep in mind.
Let's consider two series
\begin{align*}
A(x)=\sum_{n=0}^{\infty}a_nx^n\qquad\qquad B(x)=\sum_{n=0}^{\infty}b_n\frac{x^n}{n!}
\end{align*}
Two relations which are sometimes useful and which can also be found in the cited book are
\begin{align*}
&\left(a_n\right)_{n\geq 0}\quad\to\quad \left(\sum_{k=0}^{n}a_k\right)_{n\geq 0}&&A(x)\quad\to\quad \frac{1}{1-x}A(x)\\
&\left(b_n\right)_{n\geq 0}\quad\to\quad \left(\sum_{k=0}^{n}\binom{n}{k}b_k\right)_{n\geq 0}&&B(x)\quad\to\quad e^xB(x)\\
\end{align*}
