# 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?

Jul 4, 2017

I won't go into a full explanation as it too complex. But essentially:

Sum of the reciprocals

${\sum}_{r = 1}^{n} \setminus \frac{1}{r} = {H}_{n}$

Where ${H}_{n}$ is the $n t h$ harmonic number .

Sum of the reciprocals of the squares

${\sum}_{r = 1}^{n} \setminus \frac{1}{r} ^ 2 = {\pi}^{2} / 6 - {\sum}_{r = 1}^{n} \setminus \frac{\beta \left(k , n + 1\right)}{k}$

Where $\beta \left(x , y\right)$ is the Beta Function .