Help with alternating series sum

Abyssnight
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Homework Statement



Given the following: 1 - e + e2/2! - e3/3! + e4/4! + ...
Find the sum of series

Homework Equations



The MacLaurin equation for ex

The Attempt at a Solution



Well I thought that it would look like \sum(-1)^n\frac{e^n}{n!}
Tried the Ration Test and got no where. So I'm just kind of stumped
 
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Consider the MacLaurin series for ex.

ex = 1 + x + x2/2! + x3/3! + x4/4! ...

Now, what do you think you need to put into x to get the alternate series in your question?
 
Wow, haha. I must have had a long night to for some reason miss the obvious. x = -e and it works. Thank you haha.
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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