Author: Tinku Tara

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}}…

Question Number 200861 by sonukgindia last updated on 25/Nov/23 Answered by Mathspace last updated on 25/Nov/23 $${J}=_{{x}={t}^{\frac{\mathrm{1}}{\mathrm{10}}} } \frac{\mathrm{1}}{\mathrm{10}}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{5}}{\mathrm{1}+{t}}{t}^{\frac{\mathrm{1}}{\mathrm{10}}−\mathrm{1}} {dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\infty}…

Question Number 200863 by sonukgindia last updated on 25/Nov/23 Answered by Mathspace last updated on 25/Nov/23 $${x}^{\mathrm{2}\pi} ={t}\:\Rightarrow{x}={t}^{\frac{\mathrm{1}}{\mathrm{2}\pi}} \:{and} \\ $$$${I}=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{x}^{\frac{\pi}{\mathrm{5}}−\mathrm{1}} }{\mathrm{1}+{x}^{\mathrm{2}\pi} }{dx}=\frac{\mathrm{1}}{\mathrm{2}\pi}\int_{\mathrm{0}}…

Question Number 200859 by liuxinnan last updated on 25/Nov/23 $$\sqrt[{\mathrm{0}}]{{x}}=? \\ $$ Commented by mr W last updated on 25/Nov/23 $${not}\:{defined},\:{just}\:{like}\:\frac{\mathrm{1}}{\mathrm{0}}. \\ $$ Terms of…