## On the Remarkable Fact that a Sequence with Convergent Sum when Dotted with the Harmonic Sequence Yields a New Convergent Sum

December 3rd, 2017
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This is kind of incredible: take a sequence with convergent sum

and let

It turns out that

A corollary of this is that if we define

then

With this we can prove that the p-series for p>2 converges. Take the known fact that

Then

Clearly, repeated application of yields:

Next define

Since

diverges, it seems clear that as a function (say, ) is not surjective.

This clears up the question I had about whether a sequence with convergent sum dot was convergent (answer: not generally).

Let me know if you are interested in a proof (which does not rely on the Comparison Test).

Categories: Arithmetic, Functional Analysis, Infinite Sums, Mathematics