We have to find the value of $$\sum_{n=1}^{\infty} \frac{1}{n} \sum_{k=n+1}^{\infty} \frac{(-1)^k}{(2k-1)^2} + 2\int_0^{\pi/4} \log^2(\cos x) dx$$
I have no idea where to even begin with, any help/hints would be a massive help as I've been trying this one for a long time now.
I have done near about nothing in this problem except writing it down and thinking, I did think that this might be somehow related to taylor series but I have no idea whatsoever how to implement it. My first instinct was to solve the inner summation to get a function in n and then apply reimann sums but I am not able to proceed even the first step.
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maths is fun
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3What have you tried? – DecarbonatedOdes Nov 05 '22 at 21:16
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Let $k:=k+n+1$... – Bob Dobbs Nov 05 '22 at 21:25
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I tried to do some basic stuff like writing it down for a couple of iterations but am not able to proceed – maths is fun Nov 05 '22 at 21:29
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2May you show what you tried so we can help? – Accelerator Nov 05 '22 at 21:43
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For the integral, you might try using Fourier series (see here) – user170231 Nov 06 '22 at 00:05
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1I wouldn't ask about the question until you are able to articulate your thoughts into the query. Since you have some ideas, please edit your query so that you can attract more experienced users. – Accelerator Nov 06 '22 at 20:10
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You have a trigamma infinite sum – Тyma Gaidash Nov 06 '22 at 20:43