An occult sum

Calculus Level 5

\[\sum _{ n=1 }^{ \infty }{ \frac { n }{ n! } } \cos { \left( \frac { \pi n }{ 5 } \right) }\]

If the value of the above sum can be expressed in the form \[\sqrt { { e }^{ \varphi } } \cos { \left( \frac { \beta \pi +\gamma \sqrt { \alpha \left( \gamma -\sqrt { \gamma } \right) } }{ \delta } \right) }\] with \(\alpha\), \(\beta\), and \(\gamma\) positive divisors of \(\delta\); \(\alpha\) a divisor of \(\beta\); and \(\alpha\) and \(\gamma\) prime numbers, find \(\dfrac { \beta \times \delta }{ \alpha \times \gamma }\).

Notations:

  • \(e\) denotes Euler's number, the base of the natural logarithm.

  • \(\varphi\) represents the golden ratio, or \(\dfrac { 1+\sqrt { 5 } }{ 2 }\).


This problem is original.
Image Credit: Pentacle 1 by Nyo, Wikipedia.
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