0

Leonhard Euler had considered the series

$$ \sum_{i=2}^{\infty}(-1)^{n}\log i$$

and erroneously concluded that it converges to $\frac{1}{2}\log \left( \frac{\pi}{2} \right)$.

It seems plausible that he could have considered Wallis' Product

$$ \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \ldots = \frac{\pi}{2}$$

and tried to take the square root of both sides before using properties of logarithms.

However, I have not been able to find anything to substantiate this.

(1) Is anyone familiar with this problem that can shed some light on whether or not Euler may have made the mistake of trying to extend a property of finite products to an infinite one?

(2) Can anyone cite similar mistakes made by other famous mathematicians?

DDS
  • 3,199
  • 1
  • 8
  • 31
  • Euler also concluded that $1 + 3 + 9 + 27 + 81 + 243 + · · ·=-\frac12$. There was nothing erroneous about it, in his time the modern definition of the sum of a series did not exist, and his definition was that it is given by the analytic expression associated with the series. We do have $1+x+x^2+...=\frac1{1-x}$. Euler might have treated your example the same way, it doesn't mean he concluded it "converges". Is there a reference? He did treat things like $\log i$ as multivalued though, see Bal's paper. – Conifold Jul 20 '19 at 07:10
  • A related question https://math.stackexchange.com/q/3297565/269624 – Yuriy S Jul 20 '19 at 07:11
  • You may find Nahin's book on Euler interesting. On p.38 the author states that Wallis' formua is more easily obtained from Euler's own product formula for the sine function. There's lots of info in that book on what Euler did back then. Rigor was not the order of the day. Also check out Leonhard Euler and The Bernoullis by Tent, An Imaginary Tale: The Story of $\sqrt{-1}$, and The Calculus Gallery. – pshmath0 Jul 20 '19 at 07:15
  • Those regularizations don't make however a divergent sum convergent. So, an equation like the one in the first comment should not be written down this way because it seems then that the sum actually has that value. This led to huge confusion (especially the classic "equation" $1+2+3+\cdots =-1/12$). Only if it is explicitely pointed out that the extension is not allowed (since the sum does not converge at the point), but that the sum WOULD have the value IF we could use the extension, then, we can consider it to be a valid statement. – Peter Jul 20 '19 at 08:07
  • (continued) Should another extension lead to the same sum and give another value, we moreover arrive at the dilemma which value we should assign. – Peter Jul 20 '19 at 08:09

0 Answers0