S=sum of (-1)^n/n from 1 to infty
For why I named the post as so, here's why
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Good one. Thanks for posting!spoiler
ln(2) because the Taylor series for ln(1+x) evaluated at x=1 is exactly this series.
this post was submitted on 02 Jun 2024
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yup thats the intended solution, im not really familiar with taylor series yet, but maybe for a person who knows taylor series would be able to see it right away
Hint
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The solution I have in mind is related to the Taylor series
Hint 2
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It converges to -ln(2), but why
Solution:
There closest I was able to do on the fly was to show that the ::: pairwise sum has an integral of -ln(2)/2 ::: but that was pretty hand wavy. I'll have to check how it's actually done.
minus one plus a half minus a third plus a quarter... this converges to a value somewhere between 0 and -1, but I dont know how to get more precision without just running this out by hand a bunch more.