/sci/ - Science & Math

>>14950659
I meant that each term gets smaller as n gets bigger, in the harmonic series the second term is 1/2 and as n gets bigger it stays 1/2 and doesn't change, this is not true for your sum.

Anyway, I'm pretty sure you can show its 0 with some inequality like this
[eqn] x_n = \frac{1}{n^2+1} + \frac{1}{n^2+2} + \dotsb + \frac{1}{n^2+ \lfloor na \rfloor} \leq \frac{1}{n^2+1} + \frac{1}{n^2+1} + \dotsb + \frac{1}{n^2+1} = \frac{n}{n^2+1} \\ \lim_{n \rightarrow \infty} \frac{n}{n^2+1} = 0 \geq \lim_{n \rightarrow \infty} x_n [/eqn]
Since the this is a sum of non-negative terms then the original sum must be 0.