How-to: The Bernoulli numbers and Faulhaber's formula

By Terrence P. Hui, Ph.D.
In this video, we will introduce you to the Bernoulli numbers, members of an important sequence of rational numbers. We will show you the key properties of this sequence. We will also show you how they are related to the well-known Faulhaber's formula for determining 1^k+2^k+...+n^k. This is part of the playlist • Intermediate to advanc...
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[@leejongtong]

Very interesting and helpful for me to understand the details of the Bernoulli numbers. Thanks.

[@TerrencePHui]

Thanks!

[@hansulrichkeller6651]

Nicely done - your final formula is correct. The introduction of (-1)^j remains somehow strange all the same. Jakob Bernoulli by the way separated his formula for S(n,k) into an addend n^(k+1)/(n+1), then and a second addend n^k/2, and then only a SUM, from j = 2 (!) till k of ..., without the need of this (-1)^j.🥰

[@pauselab5569]

Mathologer did a great video on this as well.

[@nin10dorox]

Did Euler know that his definition of the Bernoulli numbers worked for 1^k + 2^k + ... + n^k for every k? Or did he just see thea the numbers seemed to line up, with no proof?

[@kobethebeefinmathworld953]

This is truly amazing!

[@TerrencePHui]

Thanks!