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I am having a hard time to calculate the sum of $$\sum_{n\geq 0} \frac{\cos(n)}{n!}$$ I don't know how to approach this. Could you give any thints for this?

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    $\begingroup$ $\cos n = \operatorname{Re} ( \cos n + i\sin n)$ $\endgroup$ – Daniel Fischer Oct 25 '16 at 15:50
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    $\begingroup$ I'm surprised you're dealing with $ \cos n $ for integer $ n $ - usually arguments to cosine are either continuous or integer multiples of $ \pi $ - it's unusual, say, to run into $ \sin 1 $. Is this really the problem you meant to ask? $\endgroup$ – QuantumFool Oct 25 '16 at 15:51
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    $\begingroup$ @QuantumFool, expressions such as $\sin n$ and $\cos n$ for $n \in \Bbb{Z}$ aren't that uncommon when dealing with infinite series in an academic setting. $\endgroup$ – tilper Oct 25 '16 at 15:57
  • $\begingroup$ @QuantumFool, expressions such as $\sin n$ and $\cos n$ for $n \in \Bbb{Z}$ aren't that uncommon when dealing with infinite series in an academic setting. $\endgroup$ – tilper Oct 25 '16 at 15:57
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\begin{align} \sum\frac{\cos n}{n!}& =\Re\sum\frac{e^{in}}{n!} \\ & = \Re e^{e^i}\\ & = e^{\cos 1} \cos(\sin 1) \end{align}

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    $\begingroup$ nice solution! +1) $\endgroup$ – haqnatural Oct 25 '16 at 15:59
  • $\begingroup$ Short and clear! Perfect, thank you! $\endgroup$ – John Mayne Oct 25 '16 at 17:19

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