<s>[INST] Finding the sum of this alternating series with factorial denominator. What is the sum of this series?
$$ 1 - \frac{2}{1!} + \frac{3}{2!} - \frac{4}{3!} + \frac{5}{4!} - \frac{6}{5!} + \dots $$
 [/INST] Alternatively, write it as:
$$1-\frac{1}{1!} +\frac{1}{2!} - \frac{1}{3!}... +\\
\left(-\frac{1}{1!}+ \frac{2}{2!} - \frac{3}{3!}...\right)$$
The first line is $e^{-1}$ and the second line, after cancelling terms, you see is $-e^{-1}$
More generally, if $$(z)_i = z(z-1)...(z-(i-1))$$ is the falling factorial, and $p(z) = a_0(z)_0 + a_1(a)_1 + ... a_k(z)_k$, then:
$$\sum_{n=0}^\infty \frac{p(n)}{n!} x^n = 
e^x (a_0 + a_1x + a_2x^2 + ... a_k x^k)$$
In this case, $p(z) = 1 + z = (z)_0 + (z)_1$ so
$$\sum_{n=0}^\infty \frac{n+1}{n!} x^n =
e^x (1+x)$$
And, in particular, for $x=-1$, $$\sum_{n=0}^\infty \frac{(-1)^n(n+1)}{n!} = 0$$
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