Beautiful math thread

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>>8505281
It's not much, but.. as one studies the derivative the gamma function, one finds the riemann zeta function in the serie expansion.

>>8505281
>babby's first Stirling's formula
truly trivial

>>8505281
Where does one even start? I'm an engineer and I can only think of using sterling's approximation of factorials and demonstrating that the numerator approaches the denominator?

>>8505306
Euler-Maclaurin formula applied on the sum of ln(k) from k=1 to n gives everything save for the factor squareroot of 2*pi. This one can be found from wallis integrals. Another method is to take the gamma function and by a few substitution tricks getting the result.

the square root of pi comes from gamma(1/2) which is equal to square root of pi.

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>>8505306
Just resuming what the exam was

I. integrate Wallis Integrals (pic related) and find their value.

II. Show that Wallis Integrals values are decreasing when p is increasing. Use that to demonstrate that 2n*In/pi = 1 when n -> infinite.

(In is the Wallis Integrals with p = n)

III. With the helps of limited developments and series, show that the Stirling limit exists.

IV. Combine everything.

I loved doing that mock exam.

>>8505320
Messed up p and n, lel