The way I rewrote the integral had a fatal algebraic mistake.
Thanks Mr. F
What I wrote would give
B(n,n) = G(n)^2/G(2n) = (n!)^2/(2n)! ==> the integral is (n!)^2/2(2n)! ==> the exercise gives the infinite series of
n(n!)^2/(2n)! and using the quotient test we get a_(n+1)/a_n --> 1/4 ==> the series converges.
About its value I'll pass for now.
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So the job is to calculate .
now swap integral and series, compute the series and finally compute the remaining integral.
in fact, we could've done this immediately given the problem, just swap series and integral and then compute the series. (Find a formula for and then solve the remaining integral.)