Nevermind, I noticed that if I let s = m - n, then take the limit as s -> 0, the first term does evaluate to L while the second term disappears and the answers agree. 


Nevermind, I noticed that if I let s = m - n, then take the limit as s -&gt; 0, the first term does evaluate to L while the second term disappears and the answers agree. 


https://i.imgur.com/ouYeJdG.png

I must be making a mistake somewhere because I'm getting contradictory results as far as I can tell.

If I let m = n and then integrate, I get the expected result that the integral is equal to L.

However, as far as I can tell, if I integrate without specifying m or n, I get to the last line, where all the sine terms evaluate to zero and I get the added bonus that the first coefficient blows up to infinity if I let m = n.

Can anyone see where I'm going wrong here?

Calculating an integral related to Fourier series