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On a Rather Surprising Result

April 14th, 2013 No comments

Hmm... I have figured out something rather surprising.  It is this:

Claim. The infinite sum

\lim_{n \to \infty} s^o(n) = \sum_{i=1}^\infty \frac{1}{(2i)!} = \frac{(e-1)^2}{2e}

 On the other hand the infinite sum

\lim_{n \to \infty} s^e(n) = \sum_{i=0}^\infty \frac{1}{(2i + 1)!} = \frac{e^2 - 1}{2e}

 Thus, both infinite sums are convergent.

One way to prove this is by using hyperbolic sine and hyperbolic cosine Maclaurin expansions.  But I argued it differently using (function) eigenvalues.

The proof is detailed in version 11 of "Compendium...", but since there are some ideas that are grossly incomplete (not this proof, I feel it's pretty solid) I haven't gotten around to posting it.

I'm not sure how it fits into the rather big scheme of things yet... but I'm getting there.