Category: Limits

Question Number 133997 by rs4089 last updated on 26/Feb/21 $${lim}_{n\to \mathrm{\infty }}{\left(\frac{1+\frac{1}{2}+\frac{1}{3}+\dots +\frac{1}{n}}{n^2}\right)}^n$$ Answered by mathmax by abdo last updated on 27/Feb/21 $$\mathrm{U}_{\mathrm{n}} =\left(\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{3}}+….+\frac{\mathrm{1}}{\mathrm{n}}}{\mathrm{n}^{\mathrm{2}}…

Question Number 133923 by bemath last updated on 25/Feb/21 $$\underset{x\to 0}{\lim }\frac{\mid \sin \mathrm{x}\mid }{{\mathrm{x}}^2}=?$$ Answered by EDWIN88 last updated on 25/Feb/21 $$\:\mathrm{x}^{\mathrm{2}} \:=\:\mid\mathrm{x}\mid^{\mathrm{2}} \:\Rightarrow\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mid\mathrm{sin}\:\mathrm{x}\mid}{\mid\mathrm{x}\mid}\:.\frac{\mathrm{1}}{\mid\mathrm{x}\mid}=\:\underset{{x}\rightarrow\mathrm{0}}…

Question Number 2806 by prakash jain last updated on 27/Nov/15 $$\mathrm{Prove}\mathrm{that}$$$$\underset{i=1}{\sum ^{\mathrm{\infty }}}\frac{\cos ix}{i^2}\mathrm{is}\mathrm{uniformly}\mathrm{convergent}\mathrm{on}\mathrm{real}\mathrm{line}.$$ Answered by Yozzi last updated on 27/Nov/15…

Question Number 133840 by Algoritm last updated on 24/Feb/21 Answered by Dwaipayan Shikari last updated on 24/Feb/21 $$\underset{x\to 2}{\lim }\frac{x^x\mathrm{\Gamma }(x+1)-4\mathrm{\Gamma }(3x-3)}{3\mathrm{\Gamma }(x+1)-6}=\frac{x^x(logx+1)\mathrm{\Gamma }(x+1)+{\mathrm{\Gamma }}^{\prime }(x+1)x^x-12{\mathrm{\Gamma }}^{\prime }(3x-3)}{3{\mathrm{\Gamma }}^{\prime }(x+1)}$$$$=\frac{\mathrm{8}\left({log}\left(\mathrm{2}\right)+\mathrm{1}\right)+\mathrm{4}\Gamma'\left(\mathrm{3}\right)−\mathrm{12}\Gamma'\left(\mathrm{3}\right)}{\mathrm{3}\Gamma'\left(\mathrm{3}\right)}=\frac{\mathrm{8}\left({log}\left(\mathrm{2}\right)+\mathrm{1}\right)}{\mathrm{6}\left(\frac{\mathrm{3}}{\mathrm{2}}−\gamma\right)}−\frac{\mathrm{8}}{\mathrm{3}} \…

Question Number 2735 by prakash jain last updated on 25/Nov/15 $$\mathrm{Does}\mathrm{the}\mathrm{following}\mathrm{series}\mathrm{converge}?$$$$\underset{i=1}{\sum ^{\mathrm{\infty }}}\mid \frac{\sin i}{i}\mid$$ Commented by Filup last updated on 26/Nov/15 $$\mathrm{There}\:\mathrm{is}\:\mathrm{a}\:\mathrm{limit}…