Category: Relation and Functions

Question Number 68593 by Abdo msup. last updated on 14/Sep/19 $$calculate\sum _{n=1}^{\mathrm{\infty }}\frac{{(-1)}^n}{n{(2n+1)}^2}$$ Commented by Abdo msup. last updated on 06/Oct/19…

Question Number 2891 by 123456 last updated on 29/Nov/15 $$\mathrm{given}$$$$\mathrm{\Gamma }\left(z\right)=\underset{n\to +\mathrm{\infty }}{\lim }\frac{n!}{z(z+1)\cdot \cdot \cdot (z+n)}{n}^z$$$$\mathrm{proof}\mathrm{that}$$$$i)$$$$\mathrm{\Gamma }(z+1)=z\mathrm{\Gamma }\left(z\right)$$$$ii)\mathrm{wiertrass}\mathrm{definition}\mathrm{of}\mathrm{gamma}\mathrm{function}$$$$\frac{\mathrm{1}}{\Gamma\left({z}\right)}={ze}^{{z}\gamma} \underset{{m}=\mathrm{1}} {\overset{+\infty}…

Question Number 2720 by prakash jain last updated on 25/Nov/15 $$\mathrm{Analytical}\mathrm{Continuation}$$$$\mathrm{Sum}\mathrm{of}\mathrm{the}\mathrm{below}\mathrm{divergent}\mathrm{series}\mathrm{was}$$$$\mathrm{shown}\mathrm{to}\mathrm{be}\mathrm{using}\mathrm{analytical}\mathrm{continuation}.$$$$\underset{i=1}{\sum ^n}{2}^{i-1}=-1\dots \left(\mathrm{A}\right)$$$$\zeta\left(−\mathrm{1}\right)=\underset{{i}=\mathrm{0}} {\overset{\infty} {\sum}}{i}=−\frac{\mathrm{1}}{\mathrm{12}}\:\:\:\:\:\:…\left(\mathrm{B}\right) \…

Question Number 68244 by mathmax by abdo last updated on 07/Sep/19 $$findnatureoftheserie\sum _{n=1}^{\mathrm{\infty }}arctan(n+\frac{1}{n})$$ Commented by mathmax by abdo last updated on 23/Sep/19…

Question Number 68243 by mathmax by abdo last updated on 07/Sep/19 $$letf\left(x\right)=arctan(ax+1)withareal$$$$1)calculate{f}^{\left(n\right)}\left(x\right)and{f}^{\left(n\right)}\left(0\right)$$$$2)developpfatintegrserie$$$$3)calculate{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{f\left(x\right)}{x^2+4}dx$$…