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Calculus Level 5

Define In(a)=0tnatdt\displaystyle I_n(a)=\int_0^\infty \mathrm t^na^{-t}\,\mathrm{d}t for positive real number a\displaystyle a and non-negative integer n\displaystyle n.

Then find n=01In(π)\displaystyle \sum_{n=0}^{\infty} \frac{1}{I_n(\pi)} to two decimal places.

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