MHB An almost elementary differentiation problem....

AI Thread Summary
The discussion focuses on expressing the second derivative of the Eta function at x=4 in terms of the Zeta function at the same point. The Eta function is defined as an alternating series, while the Zeta function is a standard infinite series. Participants explore the relationship between these two functions, particularly how differentiation affects their values. The problem emphasizes the need for a clear mathematical connection between the derivatives of the Eta function and the Zeta function. Ultimately, the goal is to derive a formula that links these two important mathematical concepts.
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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 $$x=4$$.
 
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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?
 
Prometheus said:
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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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.
 
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