Is there a nice way to see that the series $\sum_{n=0}^\infty (-1)^n (n+1)/n!$ converges to $0$? I just did the computation numerically. Thanks
sum of the alternating series $\sum_{n=0}^\infty (-1)^n (n+1)/n!$
-
2Hint: compute $\sum_{n=1}^N (-1)^n (n+1)/n!$. To this end, write $\sum_{n=1}^N (-1)^n (n+1)/n!$ as $\sum_{n=1}^N (-1)^n \cdot n/n! + \sum_{n=1}^N (-1)^n \cdot 1/n!$ – 2012-10-11
2 Answers
$\eqalign{ & \sum\limits_{n = 0}^\infty {{{( - 1)}^n}} \frac{{n + 1}}{{n!}} = \cr & = \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}n}}{{n!}}} + \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{n!}}} = \cr & = \sum\limits_{n = 1}^\infty {\frac{{{{( - 1)}^n}}}{{\left( {n - 1} \right)!}}} + \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{n!}}} = \cr & = \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^{n + 1}}}}{{n!}}} + \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{n!}}} = - \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{n!}}} + \sum\limits_{n = 0}^\infty {\frac{{{{( - 1)}^n}}}{{n!}}} = 0 \cr} $
-
0pretty cool actually) – 2012-10-12
First notice that $e^{-x}=\sum_{n=0}^\infty \frac{(-1)^nx^n}{n!}.$ If we multiply this by $x$, we get $xe^{-x}=\sum_{n=0}^\infty \frac{(-1)^nx^{n+1}}{n!}.$ Now if we differentiate, we get $e^{-x}-xe^{-x}=\sum_{n=0}^\infty \frac{(-1)^n (n+1)x^{n}}{n!}.$ We now set $x=1$ to get $0=e^{-1}-e^{-1}=\sum_{n=0}^\infty \frac{(-1)^n (n+1)}{n!}.$