Category: Relation and Functions

Question Number 72024 by mathmax by abdo last updated on 23/Oct/19 $${calculate}\:{lim}_{{n}\rightarrow+\infty} \:\:\:\int_{\mathrm{0}} ^{{n}} \left(\mathrm{1}−\frac{{x}}{{n}}\right)^{{n}} {ln}\left(\mathrm{1}+{x}\right){dx} \\ $$ Commented by mathmax by abdo last updated…

Question Number 72021 by mathmax by abdo last updated on 23/Oct/19 $${calculate}\:{interms}\:{of}\:{n}\:\:{U}_{{n}} =\sum_{\mathrm{1}\leqslant{i}<{j}\leqslant{n}} \:{sin}\left(\frac{{i}\pi}{{n}}\right){sin}\left(\frac{{j}\pi}{{n}}\right) \\ $$ Commented by mathmax by abdo last updated on 26/Oct/19…

Question Number 72018 by mathmax by abdo last updated on 23/Oct/19 $${calculate}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$ Commented by mathmax by abdo last…

Question Number 137365 by liberty last updated on 02/Apr/21 $${Given}\:{f}\left({x}^{\mathrm{2}} +{x}\right)+\mathrm{2}{f}\left({x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}\right)=\:\mathrm{9}{x}^{\mathrm{2}} −\mathrm{15}{x} \\ $$$${find}\:{the}\:{value}\:{of}\:{f}\left(\mathrm{2017}\right). \\ $$ Answered by MJS_new last updated on 02/Apr/21 $${f}\left({t}\right)={at}+{b}…

Question Number 6267 by sanusihammed last updated on 21/Jun/16 $${Verify}\:{the}\:{convergence}\:{of}\:{the}\:{exponential}\:{series}\:\:{e}^{{x}} \\ $$$${using}\:{D}'{Alermbert}\:{ratio}\:{test}. \\ $$ Commented by Yozzii last updated on 21/Jun/16 $${u}_{{r}} =\frac{{x}^{{r}} }{{r}!}\Rightarrow\frac{{u}_{{r}+\mathrm{1}} }{{u}_{{r}}…