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I am currently studying power series and have come across a problem I am having difficulties with. I have done some looking around on the website for a similar problem but I cant find anything that doesn't employ the lim sup method, which I am unable to use yet. Any help/guidance would be greatly appreciated!

The problem is to exactly sum the following power series:

$$\sum_{n=1}^\infty \frac{(-1)^n}{(2n)!}\pi^{2n}$$

Thank you.

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Hint: $$\cos x=\sum_{n=0}^\infty \frac{(-1)^n}{(2n)!}x^{2n}$$

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  • $\begingroup$ $n=1$ ? Shouldn't it be $n=0$. $\endgroup$ – Mustafa Said May 8 '19 at 21:50
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Tip: Consider $e^{1+i\pi}+e^{1-i\pi}-2$.

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