<s>[INST] Find a sum of appropriate values of $\cos$ and $\sin$ to determine the value of a series The task is to find a sum of multiple values $\cos$ and $\sin$ to determine the value of 
$$\sum_{n=1}^\infty (-1)^n \frac{n}{(2n+2)!}$$
Since I had no clue how to approach this I consulted Wolfram|Alpha which returned this result:
$$\sum_{n=1}^\infty (-1)^n \frac{n}{(2n+2)!} = \frac{\sin(1)}{2} + \cos(1) - \cos(0)$$
So I wrote down the partial sums of the given series and $\sin(1)$ and $\cos(1)$:
$$
\qquad\qquad\quad\sum_{n=1}^\infty (-1)^n \frac{n}{(2n+2)!} = \quad - \frac{1}{4!} + \frac{2}{6!} - \frac{3}{8!} \cdots
$$
$$
\;\;\sin(1) = \sum_{n=0}^\infty (-1)^n \frac{1^{2n+1}}{(2n+1)!} = 1 - \frac{1}{3!} + \frac{1}{5!} - \frac{1}{7!} \cdots
$$
$$
\cos(1) = \sum_{n=0}^\infty (-1)^n \frac{1^{2n}}{(2n \qquad)!} = 1 - \frac{1}{2!} + \frac{1}{4!} - \frac{1}{6!} \cdots
$$
Looking at the numbers I can see that Wolfram|Alpha's result is correct: $\frac{1}{2}1 -  \frac{1}{2!} = 0$ and $\frac{1}{2}\frac{-1}{3!} + \frac{1}{4!} = \frac{1}{4!}$, so the $\cos(1)$-series is shifted by $1$ since there is no $1$ at the beginning of the given series, so it needs to be subtracted from $\cos(1)$: $-cos(0)=-1$. But how do I get here without Wolfram|Alpha?
 [/INST] No idea how Wolfram does it. But I guess you just start with power series and play around.
\begin{equation}
X=\sum_{n=1}^\infty(-1)^n\frac{n}{(2n+2)!}=-\frac{1}{4!}+\frac{2}{6!}-\frac{3}{8!}+\dots
\end{equation}
The annoying part is the numerators. Try doubling it so that the numerators "keep up" with the denominators.
\begin{equation}
2X=-\frac{2}{4!}+\frac{4}{6!}-\frac{6}{8!}+\dots
\end{equation}
Now the numerators are consistently off the denominators by 2.
\begin{equation*}
2X=(-\frac{4}{4!}+\frac{6}{6!}-\frac{8}{8!}+\dots)+2(\frac{1}{4!}+\frac{1}{6!}-\frac{1}{8!}+\dots)
\end{equation*}
That second part is pretty much $\cos(1)$. And the numerators of course cancel out in the first part.
\begin{equation}
2X=(-\frac{1}{3!}+\frac{1}{5!}-\frac{1}{7!}+\dots)+2(\cos(1)-1/2)
\end{equation}
Now the first part is pretty much $\sin(1)$.
\begin{equation}
2X=(\sin(1)-1)+2\cos(1)-1
\end{equation}
\begin{equation}
X=\frac{\sin(1)}{2}+\cos(1)-1
\end{equation}
Hope that didn't seem too random. My approach is to try to build the desired series out of ones I know using various manipulations like calculus, combining and rearranging.
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