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Archive for December, 2017

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

December 3rd, 2017 No comments

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 $H$ yields:

Next define

Since

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

This clears up the question I had about whether a sequence with convergent sum dot $H^{-1}$ was convergent (answer: not generally).

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

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