Zeta function and summation convergence

[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?

 

[Santa1]

How did you arrive at the sum?

 

[camilus]

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]

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
$$\sum_{k=2}^\infty \frac{(-1)^k \zeta(k)}{e^k}$$

But in the original zeries, the $$k=1$$ term is the problem.

 

[Santa1]

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.