To prove the convergence of the p-series

∞∑n=11np

for p>1, one typically appeals to either the Integral Test or the Cauchy Condensation Test.

I am wondering if there is a self-contained proof that this series converges which does not rely on either test.

I suspect that any proof would have to use the ideas behind one of these two tests.

**Answer**

We can bound the partial sums by multiples of themselves:

S2k+1=2k+1∑n=11np=1+k∑i=1(1(2i)p+1(2i+1)p)<1+k∑i=12(2i)p=1+21−pSk<1+21−pS2k+1.

Then solving for S2k+1 yields

S2k+1<11−21−p,

and since the sequence of partial sums is monotonically increasing and bounded from above, it converges.

(See also: Teresa Cohen & William J. Knight, *Convergence and Divergence of ∑∞n=11/np*, Mathematics Magazine, 52(3), 1979, p.178. https://doi.org/10.1080/0025570X.1979.11976778)

**Attribution***Source : Link , Question Author : admchrch , Answer Author : Zach Teitler*