Question Number 29987 by abdo imad last updated on 14/Feb/18 $${prove}\:{that}\:\:\sum_{{n}=\mathrm{1}_{{n}\neq{p}} } ^{\infty} \:\:\frac{\mathrm{1}}{{n}^{\mathrm{2}} \:−{p}^{\mathrm{2}} }\:\:=\:\:\frac{\mathrm{3}}{\mathrm{4}{p}^{\mathrm{2}} }\:. \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 29986 by abdo imad last updated on 14/Feb/18 $${find}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{{n}+\mathrm{1}}{\mathrm{4}^{{n}} }\:. \\ $$ Commented by abdo imad last updated on 14/Feb/18 $${let}\:{introduce}\:{for}\:\mid{x}\mid<\mathrm{1}\:\:\:{S}\left({x}\right)=\:\sum_{{n}=\mathrm{0}}…
Question Number 29985 by abdo imad last updated on 14/Feb/18 $${prove}\:{that}\:\:\:\:\sum_{{p}=\mathrm{1}} ^{\infty} \:\:\:\:\:\:\frac{{a}^{{p}} }{\mathrm{1}−{a}^{\mathrm{2}{p}} }\:=\:\sum_{{p}=\mathrm{1}} ^{\infty} \:\:\:\:\frac{{a}^{\mathrm{2}{p}−\mathrm{1}} }{\mathrm{1}−{a}^{\mathrm{2}{p}−\mathrm{1}} }\:. \\ $$ Terms of Service Privacy…
Question Number 29984 by abdo imad last updated on 14/Feb/18 $${prove}\:{that}\:\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{{H}_{{n}} }{{n}!}=={e}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\:\frac{\left(\mathrm{1}\right)^{\boldsymbol{{n}}−\mathrm{1}} }{\boldsymbol{{n}}\:\left(\boldsymbol{{n}}!\right)}\:. \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 29982 by abdo imad last updated on 14/Feb/18 $${let}\:{give}\:{f}\left({x}\right)=\sqrt{{x}+\sqrt{\mathrm{1}+{x}^{\mathrm{2}} \:}}\:\:\:\:{developp}\:{f}\:{at}\:{integr}\:{series} \\ $$$${in}\:{point}\:\mathrm{0} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 29983 by abdo imad last updated on 14/Feb/18 $${find}\:{radius}\:{andsum}\:{of}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{n}−\mathrm{1}}{{n}!}\:{x}^{{n}} . \\ $$ Commented by abdo imad last updated on 15/Feb/18 $${S}\left({x}\right)=\sum_{{n}=\mathrm{1}}…
Question Number 29981 by abdo imad last updated on 14/Feb/18 $${find}\:{radius}\:{and}\:{sum}\:{of}\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{\mathrm{2}{n}} }{\mathrm{2}{n}+\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{\mathrm{1}}{\left(\mathrm{2}{n}+\mathrm{1}\right)\mathrm{9}^{{n}} }\:. \\ $$ Commented by abdo imad…
Question Number 29978 by abdo imad last updated on 14/Feb/18 $${let}\:{give}\:{x}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{prove}\:{that}\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dt}}{\mathrm{1}+{t}^{{x}} }=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{nx}+\mathrm{1}} \\ $$$$\left.\mathrm{2}\right)\:{find}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+\mathrm{1}}\:\:{and}\:\sum_{{n}=\mathrm{0}} ^{\infty}…
Question Number 29979 by abdo imad last updated on 14/Feb/18 $${find}\:{the}\:{radius}\:{of}\:{S}\left({x}\right)=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{{x}^{\mathrm{3}{n}+\mathrm{2}} }{\mathrm{3}{n}+\mathrm{2}} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\:\frac{\mathrm{1}}{\left(\mathrm{3}{n}+\mathrm{2}\right)\mathrm{3}^{{n}} }. \\ $$ Commented by abdo imad…
Question Number 29973 by abdo imad last updated on 14/Feb/18 $${find}\:\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\frac{{sin}\left({n}\alpha\right)}{{n}}\:{x}^{{n}} \:{with}\:\:−\mathrm{1}<{x}<\mathrm{1}. \\ $$ Commented by abdo imad last updated on 16/Feb/18 $${let}\:{put}\:{S}\left({x}\right)=\sum_{{n}=\mathrm{1}}…