How can I find convergence of this sum, without using the integral test: (sum goes from 1 to inf)

Question

∑(sqrt(n+2) - 2sqrt(n+1) + sqrt(n))

 

Answer 1

Indeed it seems that other criteria except Integral do not work for this series, though the series is convergent.

You can use Cauchy criteria |S_n+p-S_n|<ε. Or you can just find

lim_n→∞S_n. This limit is finite, so the series convergences.