My Sixteenth Problem

Calculus Level 3

Evaluate ∑ n = − ∞ ∞ 1 ∣ n ∣ ! \displaystyle\sum_{n= -\infty}^{\infty} \frac{1}{|n|!} .

Round your answer to 3 decimal places.


The answer is 4.437.

Daniel Ellesar by Daniel Ellesar , 24, United Kingdom

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1 solution

Atul Kumar Ashish
Mar 23, 2017

∑ n = − ∞ ∞ 1 ∣ n ∣ ! = 1 + 2 ∑ n = 1 ∞ 1 n ! = 1 + 2 ( 1 + 1 2 ! + 1 3 ! + 1 4 ! + ⋯ ) = 1 + 2 ( e − 1 ) = 2 e − 1 ≈ 4.437 \begin{aligned} & \sum_{n=-\infty}^{\infty}\frac{1}{|n|!} \\ & =1+2\sum_{n=1}^{\infty}\frac{1}{n!} \\ & =1+2\left(1+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots\right) \\ & =1+2\left(e-1\right) \\ & =2e-1 ≈\boxed{4.437}\end{aligned}

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