Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 71216 by TawaTawa last updated on 13/Oct/19

Answered by mind is power last updated on 13/Oct/19

Σ((n^2 +x)/(n!))=Σ(n^2 /(n!))+xΣ(1/(n!))  Σ_(n≥0) (n^2 /(n!))=1+Σ_(n≥2) (n/((n−1)!))=Σ((n−1+1)/((n−1)!))=Σ(1/((n−2)!))+Σ(1/((n−1)!))  =e+e−1  ⇒Σ(n^2 /(n!))=1+2e−1=2e  Σ(x/(n!))=xe  ⇒2e+xe=ex^2   ⇒x^2 −x−2=0  (x−2)(x+1)=0  x∈{2,−1}

$$\Sigma\frac{\mathrm{n}^{\mathrm{2}} +\mathrm{x}}{\mathrm{n}!}=\Sigma\frac{\mathrm{n}^{\mathrm{2}} }{\mathrm{n}!}+\mathrm{x}\Sigma\frac{\mathrm{1}}{\mathrm{n}!} \\ $$$$\underset{\mathrm{n}\geqslant\mathrm{0}} {\sum}\frac{\mathrm{n}^{\mathrm{2}} }{\mathrm{n}!}=\mathrm{1}+\sum_{\mathrm{n}\geqslant\mathrm{2}} \frac{\mathrm{n}}{\left(\mathrm{n}−\mathrm{1}\right)!}=\Sigma\frac{\mathrm{n}−\mathrm{1}+\mathrm{1}}{\left(\mathrm{n}−\mathrm{1}\right)!}=\Sigma\frac{\mathrm{1}}{\left(\mathrm{n}−\mathrm{2}\right)!}+\Sigma\frac{\mathrm{1}}{\left(\mathrm{n}−\mathrm{1}\right)!} \\ $$$$=\mathrm{e}+\mathrm{e}−\mathrm{1} \\ $$$$\Rightarrow\Sigma\frac{\mathrm{n}^{\mathrm{2}} }{\mathrm{n}!}=\mathrm{1}+\mathrm{2e}−\mathrm{1}=\mathrm{2e} \\ $$$$\Sigma\frac{\mathrm{x}}{\mathrm{n}!}=\mathrm{xe} \\ $$$$\Rightarrow\mathrm{2e}+\mathrm{xe}=\mathrm{ex}^{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{x}^{\mathrm{2}} −\mathrm{x}−\mathrm{2}=\mathrm{0} \\ $$$$\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}+\mathrm{1}\right)=\mathrm{0} \\ $$$$\mathrm{x}\in\left\{\mathrm{2},−\mathrm{1}\right\} \\ $$$$ \\ $$

Commented by TawaTawa last updated on 13/Oct/19

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com