Prove that .

 

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Here is a calculus based approach.

Let . For n>1 we can integrate this by parts to get



It follows that , and by an easy induction argument .

Now use the power series expansion of the exponential to see that

.

(It's okay to switch the sum and the integral because the series converges uniformly.)

Another way to calculate the same integral is


Therefore . Now divide both sides by x and then put x = 1/2.