A mathematical journey

Calculus Level 4

\[{ a }_{ n+1 }=2n{ a }_{ n }\] The recurrence relation above has the initial condition of \(a_1=1\). Suppose we define the function \[S\left( x \right) =\sum _{ n=1 }^{ \infty }{ \frac { { x }^{ n } }{ { a }_{ n } } }.\] Find the value of \(\displaystyle \int _{ 0 }^{ 2 }{ S\left( x \right) \, dx }\).


This problem is original.

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