10c - Abel's Partial Summation Formula

In this video we look at Abel's Partial Summation Formula which helps us explore the behaviour of arithmetic functions by transforming them to continuous functions.

We derive the formula and then illustrate its power by showing how easy it is to show that the harmonic series grows like ln(n)+O(1).

Slides:

Blog:

––––––––––––––––––––––––––––––

The following two pieces of music are freely licensed for use in this video.

[laviekolchinsky]

Please continue uploading sir. I'm not sure if you'll read this but all of your latest uploads have been excellent. This is my favourite one :)

[aedilnlprd]

Sir, Thank you so much for the video, you're playing the role of my mathematics mentor.

[akrishna]

Just discovered your channel - these videos are fantastic, thanks so much!

[rogierbrussee]

The Abel summation formula is essentially partial integration Let f be a smooth function and A(x) = ∑_{n < x} a(x)  

∑_{ y <= n < x} a(n) f(n) = ∫{y-}^{x} f(t) dA(t)

where the latter is a Riemann Stieltjes integral, which you can think of the limit of the usual Riemann integral when you smooth the function A(x) and. The Stieltjes integral has partial integration just like in calculus (which follows e.g. by doing the partial integration for the smoothed version of A)

∫_{y-}^x f(t) dA(t) = f(t)A(t)|_{y-}^x - ∫A(t) df(t) = f(x)A(x) - f(y)A(y) - ∫_y^x A(t)f'(y) dy