MHB An almost elementary differentiation problem....

[DreamWeaver]

Let $$\zeta(x)$$ be the Zeta function (where, for convenience, $$x$$ is assumed to be $$> 1$$).

$$\zeta(x) = \sum_{k=1}^{\infty}\frac{1}{k^x}$$Similarly, define the Eta function (alternating Zeta function) by the following series - where again, in this case, we assume $$x > 1$$:$$\eta(x) = \sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^x}$$
Problem:Express the second derivative of the Eta function - at $$x=4$$ - in terms of the Zeta function (differentiated or otherwise), at $$x=4$$.

 

[Prometheus1]

Spoiler

Isn't it just a matter of applying the product rule on $ \displaystyle \displaystyle \eta(x) = \left(1-2^{1-x}\right) \zeta(x)$ twice?

 

[DreamWeaver]

Spoiler

Isn't it just a matter of applying the product rule on $ \displaystyle \displaystyle \eta(x) = \left(1-2^{1-x}\right) \zeta(x)$ twice?

Thanks for taking part, Prometheus! :D

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Spoiler

Yes, you're absolutely correct. The idea of the problem was to, potentially, encourage people who're less familiar with the functional relation you cited to develop it for themselves, then differentiate it.