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Question Number 200903 by MrGHK last updated on 26/Nov/23

Answered by witcher3 last updated on 26/Nov/23

$$\mathrm{Hint}$$$${(-1)}^{\mathrm{n}}(\mathrm{\Psi }(\mathrm{n}+\frac{1}{2})-\ln \left(\mathrm{n}\right))=\sum _{\mathrm{n}⩾1}(\ln \left(\frac{2\mathrm{n}+1}{2\mathrm{n}}\right)+\mathrm{\Psi }(2\mathrm{n}+\frac{1}{2})-\mathrm{\Psi }(2\mathrm{n}+\frac{3}{2}))$$$$=\underset{x\to \mathrm{\infty }}{\lim ln}\left(\underset{1}{\prod ^{\mathrm{x}}}\frac{2\mathrm{n}+1}{2\mathrm{n}}\right)-\underset{\mathrm{k}=1}{\sum ^{\mathrm{x}}}\frac{2}{4\mathrm{n}+1}$$$$$$

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