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Calculus Level 5

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

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

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