1 + 2 + 3 + ... = -1/12

[Thomas Klemm]

(10-26-2024 11:08 PM)Johnh Wrote:  But, I can't accept the conjecture that the the series 1 -1 +1 -1 +1... etc is in some way equal to 0.5. While a converging series can be seen to converge on a value, this series is never either exactly 1/2 nor anywhere near it and that's true no matter how many terms we look at! It is simply an undefined result that may have one of two values after n terms.

From Cesàro summation:

Quote:The Cesàro sum is defined as the limit, as n tends to infinity, of the sequence of arithmetic means of the first n partial sums of the series.

The two examples of this article show that the Cesàro sum of the series \(1-1+1-1+1-\cdots\) is 1/2 while \(1+2+3+4+\cdots\) is not Cesàro summable.