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These days I came across this series and I'm trying to figure out how to compute it

$$\sum_{k=0}^{\infty} \frac{3}{(3 k)!}$$

I thought to combine some elementary functions, but it doesn't work. Some hints, suggestions?

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but... but... $3/3k=1/k$ – vakufo Sep 19 '12 at 16:01
    
That's true, @vakufo , yet $\,3/3k\,$ is not what is written in that sum.... – DonAntonio Sep 19 '12 at 16:35
    
Oh my god, I see it now. – vakufo Sep 19 '12 at 16:36
up vote 7 down vote accepted

Let $\omega$ be a complex cube root of 1. Think about $$e^{\omega x}+e^{\omega^2x}+e^x$$

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I didn't think of that – user 1618033 Sep 19 '12 at 13:51

Hints:

$$\sum_{k=0}^\infty\frac{1}{k!}=e$$

$$\sum_{k=0}^\infty\frac{1}{k!}=\sum_{k=0}^\infty\left[\frac{1}{(3k)!}+\frac{1}{(3k+1)!}+\frac{1}{(3k+2)!}\right]$$

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hmmm, interesting trick – user 1618033 Sep 19 '12 at 13:49

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