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Question-200903




Question Number 200903 by MrGHK last updated on 26/Nov/23
Answered by witcher3 last updated on 26/Nov/23
Hint  (−1)^n (Ψ(n+(1/2))−ln(n))=Σ_(n≥1) (ln(((2n+1)/(2n)))+Ψ(2n+(1/2))−Ψ(2n+(3/2)))  =lim_(x→∞) ln(Π_1 ^x ((2n+1)/(2n)))−Σ_(k=1) ^x (2/(4n+1))
$$\mathrm{Hint} \\ $$$$\left(−\mathrm{1}\right)^{\mathrm{n}} \left(\Psi\left(\mathrm{n}+\frac{\mathrm{1}}{\mathrm{2}}\right)−\mathrm{ln}\left(\mathrm{n}\right)\right)=\underset{\mathrm{n}\geqslant\mathrm{1}} {\sum}\left(\mathrm{ln}\left(\frac{\mathrm{2n}+\mathrm{1}}{\mathrm{2n}}\right)+\Psi\left(\mathrm{2n}+\frac{\mathrm{1}}{\mathrm{2}}\right)−\Psi\left(\mathrm{2n}+\frac{\mathrm{3}}{\mathrm{2}}\right)\right) \\ $$$$=\underset{{x}\rightarrow\infty} {\mathrm{lim}ln}\left(\underset{\mathrm{1}} {\overset{\mathrm{x}} {\prod}}\frac{\mathrm{2n}+\mathrm{1}}{\mathrm{2n}}\right)−\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{x}} {\sum}}\frac{\mathrm{2}}{\mathrm{4n}+\mathrm{1}} \\ $$$$ \\ $$

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