Math Help - find the sum of the series

1. find the sum of the series

Find the sum of the series $\sum_{n = 1}^\infty \frac{{-1}^{n }*{4}^{n } }{ n! n}$

2. Originally Posted by jayantamath
Find the sum of the series $\sum_{n = 1}^\infty \frac{{-1}^{n }*{4}^{n } }{ n! n}$

If you mean

$\sum_{n=1}^{+\infty}\dfrac{(-1)^n4^n}{n!n}$

then, consider the functional series

$f(x)=\sum_{n=1}^{+\infty}\dfrac{(-1)^nx^n}{n!n}$

$f'(x)=\dfrac{1}{x}\sum_1^{+\infty}\dfrac{(-1)^nx^n}{n!}=\dfrac{e^{-x}-1}{x}$

You'll obtain

$f(4)=\textrm{Ei}(-4)-\gamma-\log 4$

where

$\textrm{Ei}(x)=-\int_{-x}^{+\infty}\dfrac{e^{-t}dt}{t}$

and $\gamma$ the Euler-Mascheroni constant.