An Identity Involving Bernoulli Numbers and a Proof of Euler’s Formula for Zeta(2n)

Algebra, Number theory, Real analysis

Abstract: Let B_n be the sequence of Bernoulli numbers, with B_1=1/2. In this note, we derive an identity involving Bernoulli numbers. Using that, we give a proof of Euler’s formula for \zeta(2n)=\sum_{k=1}^{\infty} k^{-2n}, n\ge 1.

The note is here: Zeta(2n).

I wrote this about two years ago (when I just graduated from high school). But due to lack of experience, that note is terrible (to read), so I made a new version of it.

Edit: Later I found that it is essentially same as Apostol’s proof given in [1]. So its nothing new, though as a high school student I really enjoyed deriving it on my own!

Reference:
[1] Apostol, T. (1973). Another Elementary Proof of Euler’s Formula for ζ(2n). The American Mathematical Monthly, 80(4), 425-431. doi:10.2307/2319093

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