An integral from Putnam 1969
Before looking at the solution, give the problem a go!
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Let us firstly recall that for real x, the following identity holds:
hence we may rewrite our integral as:
Now, if we recall the well-known power series for exp(x), we get that this integral is nothing but
However, due to uniform convergence, we may swap the order of the integral and sum to give us:
This integral begs for an exponential substitution, especially that it looks like it could correspond to the Gamma function. If we attempt to substitute x = exp(-u) we get the following:
This is ALMOST in the form of the Gamma function, the only obstacle in the way now is this n+1 part of the exponential. However, this is simple to solve by simply performing a substitution of the form t = (n+1)u to finally yield:
Now, pulling everything back together, we get the following, given that this final integral evaluates to n! from the gamma function:
Hence,
Which we can write nicely as:
