Zeta function and summation convergence

**rman144** · Jun27-09, 02:50 PM

I need to know if the following series converges:

∑(k=1 to k=oo)[(((-1)^k) ζ(k))/(e^k)]

The problem is that zeta(1)=oo; however, the equation satisfies the conditions of convergence for an alternating series [the limit as k->oo=0 and each term is smaller than the last.]

Any thoughts?

*-<|:-D=<-< · Jun27-09, 03:52 PM

How did you arrive at the sum?

**camilus** · Jun27-09, 05:53 PM

well yeah I see what you're saying about zeta of 1. To see if the summation converges, try one of the tests, like tha ratio test.

**g_edgar** · Jun28-09, 08:01 AM

Originally Posted by rman144

I need to know if the following series converges:

∑(k=1 to k=oo)[(((-1)^k) ζ(k))/(e^k)]

The problem is that zeta(1)=oo; however, the equation satisfies the conditions of convergence for an alternating series [the limit as k->oo=0 and each term is smaller than the last.]

Any thoughts?

This one converges
$LaTeX Code: <BR>\\sum_{k=2}^\\infty \\frac{(-1)^k \\zeta(k)}{e^k}<BR>$

But in the original zeries, the

term is the problem.

*-<|:-D=<-< · Jun28-09, 04:18 PM

Yes if memory serves me right that sum is just a constant and an x away from being a taylor series of the digamma function.