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Series of \ast should be treated as a unit
I’m doing some exercises on this website.
I am trying to evaluate:
$$\sum _{ k=1 }^\infty \frac { (-1)^{k+1} } { k! \left( k+1\right) }$$
I notice that \ast acts as a unit when working with integers. However, if I use it for the whole expression above the \sum, only the first term of the expansion is correct.
For instance, I obtain
$$\sum _{ k=1 }^\infty \frac { (-1)^{k+1} } { k! \left( k+1\right) }=1-1=0$$
Where is my error?
A:
Using the double-angle method,
$$
\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k!(k+1)}=\sum_{k=1}^\infty\left(\frac1{k!k}-\frac1{k!(k+1)}\right)
$$
Now use the fact that
$$
\frac{k!}k=\frac{(k-1)!}1+\frac{k!}k
$$
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