To prove the convergence of the p-series
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.
We can bound the partial sums by multiples of themselves:
Then solving for S2k+1 yields
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)