Question Number 200903 by MrGHK last updated on 26/Nov/23
Answered by witcher3 last updated on 26/Nov/23
$$\mathrm{Hint} \\ $$$$\left(−\mathrm{1}\right)^{\mathrm{n}} \left(\Psi\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}\right)−\mathrm{ln}\left(\mathrm{n}\right)\right)=\underset{\mathrm{n}\geqslant\mathrm{1}} {\sum}\left(\mathrm{ln}\left(\frac{\mathrm{2n}+\mathrm{1}}{\mathrm{2n}}\right)+\Psi\left(\mathrm{2n}+\frac{\mathrm{1}}{\mathrm{2}}\right)−\Psi\left(\mathrm{2n}+\frac{\mathrm{3}}{\mathrm{2}}\right)\right) \\ $$$$=\underset{{x}\rightarrow\infty} {\mathrm{lim}ln}\left(\underset{\mathrm{1}} {\overset{\mathrm{x}} {\prod}}\frac{\mathrm{2n}+\mathrm{1}}{\mathrm{2n}}\right)−\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{x}} {\sum}}\frac{\mathrm{2}}{\mathrm{4n}+\mathrm{1}} \\ $$$$ \\ $$