# 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 $$\dfrac { 1+\sqrt { 5 } }{ 2 }$$.

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