Do these infinite series converge to a finite limit?

Question

Now, I know that there is that remarkable result which finds that $$\sum_{n=1}^{\infty}n=-\frac{1}{12}$$ for $n\in\mathbb{N}$, under some kind of Cauchy limit.

Are there any such convergences for the following series summations?

$\sum_{n=0}^{\infty} n$;
$\sum_{n=0}^{\infty} n^2$.

You could call these summability methods, but it's certainly not a case of convergence. — Robert Israel, Jan 21 '16 at 06:14
Good point, @RobertIsrael -- I suppose a more accurate term would be conditional convergence, no? — Harry Smith, Jan 21 '16 at 06:22
@HarrySmith no, conditional convergence still refers to convergent series (which are just more fragile than absolutely convergent series). These above sums are actually divergent - you can just assign a value to them by somehow writing it as "something" + the infinity we don't care about. — orion, Jan 21 '16 at 06:25

score 2 · Accepted Answer · answered Jan 21 '16 at 06:12

These are regularized sums. The first one is actually the one that you already wrote in the first line (adding zero doesn't do anything). The second turns out to regularize to zero. Read up on zeta regularization and watch these excellent videos

https://www.youtube.com/watch?v=w-I6XTVZXww

https://www.youtube.com/watch?v=0Oazb7IWzbA

And you can generalize $\sum_{n=0}^\infty n^k = \zeta(-k)$ where $\zeta$ is the Riemann zeta function - you can see from the definition of this function, that all we do is to put in a number outside the convergence domain of original series, so the result is not a true sum of the series (which diverges) but analytic continuation that extends the domain beyond what the series can do.

Do these infinite series converge to a finite limit?

1 Answers1