Let be a convergent series of nonnegative terms (i.e., for all ). Prove that
converges for all . Provide an example to show this fails in the case .
(Note. Thanks to Giovanni in the comments for pointing out the problems with the original proof.)
Proof. We apply the Cauchy-Schwarz inequality. (See this exercise for some more proofs regarding the Cauchy-Schwarz inequality.) The Cauchy-Schwarz inequality establishes that
So, to apply this to the current problem, we let
Then, since we have ; hence, . So, we have
But, on the right we know converges (by hypothesis) and we know converges for . Hence, each of the sums on the right converges, so the term on the right is some finite number . This means the partial sums of are bounded by a constant . By Theorem 10.7 (page 395 of Apostol) this implies the convergence of the series
Example. Let and let . Then, we know converges (by Example #2 on page 398 of Apostol, established with the integral test). However,
diverges by the same example on page 398. Therefore, we conclude the theorem fails in the case .