Integration, Generalization and Sum

In general for $n \in\mathbb Z \geq 0$ , $I_n =\int_{-\infty}^{\infty} \dfrac{1}{\,(x^2+1)^{n+1}} = \dfrac{\,(2n)!}{\,(2^n n!)^2}\pi$ Then $\sum_{n=0}^{\infty} \frac{\,(2n)!!}{\,(2n)!}I_n$ can be expressed as $\pi^A \times e^B$ , where $A$ and $B$ are rational numbers.

Submit your answer as the value of the fraction $\frac{A}{B}$ .

Notations:

• $\lfloor \cdot \rfloor$ denotes the floor function .

• $e \approx 2.71828$ denotes the Euler's number .

Inspiration .