What is the sum of the series 1/1, 1/2, 1/3, ... 1/n and 1/1, 1/4, 1/9....1/n^2 where n is finite?
I won't go into a full explanation as it too complex. But essentially:
Sum of the reciprocals
# sum_(r=1)^n \ 1/r = H_n#
Sum of the reciprocals of the squares
# sum_(r=1)^n \ 1/r^2 = pi^2/6 - sum_(r=1)^n \ (beta(k,n+1))/k#