The alternating harmonic series

[@naribaek5289]

I love this presentation of the material. Can I ask what software you are using? It looks like PowerPoint, but I'm not sure.

[@chrisodden]

Hi, Nari. I use Keynote on a mac. For mathematical expressions I use LaTeXit, one of the apps that comes bundled in the MacTeX package.

[@naribaek5289]

@@chrisodden Thank you so much for the info! :)

[@prognosticii]

At 7:40, your steps to prove both subsequences converge to the same limiting value are a little confusing: where did 1/(2n+1) come from? I can prove the expression is true with test values of n, but is there a more formal derivation?

[@chrisodden]

Hi Trey. (I will use s for "sigma".) The partial sum s_N is simply partial sum a_1 + a_2 + ... + a_N. Whenever we subtract one partial sum from the previous partial sum we simply obtain the single term that was added to get from the one sum to the next. That is, s_{N+1} - s_N = (a_1 + ... + a_N + a_{N+1} ) - (a_1 + ... + a_N ) = a_{N+1}. In the video the application of this principle is: s_{2N+1} - s_{2N} = a_{2N+1} = 1/(2N+1).

[@chrisodden]

I might add that for the alternating harmonic series a_k = (-1)^(k+1) / k, so using k = 2N+1 means that a_{2N+1} = 1/(2N+1).

[@prognosticii]

@Chris Odden I see. It was just odd for me, as when I derived the proof that the even terms increase that constant added was different from the one for the odd claim, and hence my initial reservation to your deduction.

[@anaboumard9482]

You deserve way more subscribers !! I can't find the exact proof of why this series converges to ln2 tho...

[@chrisodden]

Thanks! As for the value of the series, the usual proof is to show that the values of the function ln(1+x) are given by the power series x - x^2/2 + x^3/3 - x^4/4 + ... (known as the Taylor series for ln(1_x)) for each x in the open interval (-1,1), then to appeal to Abel's Theorem, allowing x to approach 1 from the left to obtain ln 2 = 1 - 1/2 + 1/3 - 1/4 + ...

[@cmPe6an]

@@chrisodden I have seen the proof done using ln(1-x) and then letting x approach -1. I have not seen Abel's Theorem cited in those proofs. Is it being used but swept under the rug? Maybe when you are sure that the series is going to converge at the endpoint, you can just plug it in and say "look it's the alternating harmonic series so it converges" instead of citing Abel's Theorem first. Thanks for the video!

[@prognosticii]

@@cmPe6an I watched a video from BriTheMathGuy on the proof of the convergence, though it is not the most intuitive; but, it is the proof irregardless. The proof evaded use of taylor series, &c.

[@comic4relief]

I graphed it with a piece of paper and a pencil.

[@chrisodden]

Yes! I should have mentioned that - two of my favorite technologies, pencil and paper.

[@comic4relief]

@@chrisodden ...and eraser!