Is it true that Euler did not prove the sum of the Basel series $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$?

Question

I've heard that Euler actually never managed to prove $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$ If this is true, then where does the common misconception come from?

Answer 1

The question can be answered "yes" or "no" depending on how the word "prove" is interpreted. Euler was working in the context of "algebraic analysis" of 18th century, so he wasn't primarily concerned with issues that underlie modern definition of convergence, despite performing many numerical computations. Historian Ferraro writes:

"The Eulerian definition of the sum of a series should be understood in such a context. According to this definition, the value of the sum of a series was the value of the function from the development of which the series was generated."

For example, according to Euler, $$1+2+2^2+\dots=\frac{1}{1-2}=-1,$$ by definition! Deciding whether his proofs are "rigorous" by modern standards is like fitting square peg into round hole, he operated in a different environment and under different rules.

Euler gave several derivations for the sum of the Basel series. The first one, where he manipulated infinite series as if they were polynomials to factorize $1-\sin s$, naturally "is not considered valid today". However, some of its intuitions can be justified using non-standard analysis, see Kalman's Six Ways to Sum a Series. At the time it did not satisfy even Euler himself as it raised some serious questions. How can we be sure that $1−\sin s$ does not have other roots besides the obvious ones? How come $\sin s$ and $e^s\sin s$ have the same roots, but presumably different factorizations?

"It took him about ten years, but he finally succeeded in obtaining the famous product formula for $\sin s$... The proof... was beautiful and direct."

But... it still involved formal manipulation of complex numbers, poorly understood at the time, and was missing justifications for uniform convergence, absolute convergence, or even a notion of convergence for that matter. So it wasn't "rigorous" by modern standards either.

Cauchy gave a more elementary proof in Cours d'Analyse (1821), which is also not beyond modern reproach. Euler's "gaps" were eventually filled by Weierstrass, but they were what is called "technical", see Varadarajan's Euler and his Work on Infinite Series for details.

Answer 2

No it's not.

I think the misconception comes from the fact that Euler announced the result five years before providing a correct proof (Euler, Opera Omnia Series 1, Volume 14, p. 177-186 (E63)). See this webpage for the original proof Euler used (the 1735 version, not the 1741 amended version).

Euler is well-known for unjustified mathematics manipulations (think of the sum of the alternate series being worth $1/2$), and his first "proof" relied on extending properties of the product expression of polynomials to infinite polynomial series. It took him 10 years, and development in the understanding of series, to show a second version, that was less litigious, hence the misconception you're talking about: between publishing his discovery in 1735 and proving it rigorously in 1741.

However, "proof" clearly has a different meaning today than in Euler's mind, as Conifold’s answer addresses.

Note that Cauchy provided a simpler, more elegant proof in 1820.

Answer 3

It really depends on what you think constitutes a 'proof'. Euler gave a proof for $\zeta(2)=\pi^2/6$ with an argument that $\sin(\pi x)/\pi x$ should have a factorization like a polynomial in terms of its (infinitely many) roots. This would be convincing to most calculus students. He gave the values of $\zeta(2n)$ by manipulating series expansions of $\cot(z)$. This would be convincing to undergraduate math majors. In both cases there are certainly subtleties that need to be considered. For example, in his general proof there are double limits that are interchanged formally, and this needs justification.