Question and Answers Forum

All Questions      Topic List

Relation and Functions Questions

Previous in All Question      Next in All Question      

Previous in Relation and Functions      Next in Relation and Functions      

Question Number 31983 by abdo imad last updated on 17/Mar/18

calculate Σ_(n=0) ^∞ ((n^2 −2)/(n!)) .

$${calculate}\:\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{n}^{\mathrm{2}} \:−\mathrm{2}}{{n}!}\:\:. \\ $$

Commented by prakash jain last updated on 18/Mar/18

n^2 −2=an(n−1)+bn+c c=−2 b=1 a=1 n^2 −2=n(n−1)+n−2 Σ_(n=0) ^∞ ((n^2 −2)/(n!))=Σ_(n=0) ^∞ (((n(n−1))/(n!))+(n/(n!))−(2/(n!))) =e+e−2e=0

$${n}^{\mathrm{2}} −\mathrm{2}={an}\left({n}−\mathrm{1}\right)+{bn}+{c} \\ $$$${c}=−\mathrm{2} \\ $$$${b}=\mathrm{1} \\ $$$${a}=\mathrm{1} \\ $$$${n}^{\mathrm{2}} −\mathrm{2}={n}\left({n}−\mathrm{1}\right)+{n}−\mathrm{2} \\ $$$$\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{2}} −\mathrm{2}}{{n}!}=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left(\frac{{n}\left({n}−\mathrm{1}\right)}{{n}!}+\frac{{n}}{{n}!}−\frac{\mathrm{2}}{{n}!}\right) \\ $$$$={e}+{e}−\mathrm{2}{e}=\mathrm{0} \\ $$

Commented by abdo imad last updated on 18/Mar/18

Σ_(n=0) ^∞ ((n^2 −2)/(n!)) =Σ_(n=0) ^∞ (n^2 /(n!)) −2 Σ_(n=0) ^∞ (1/(n!)) but we have Σ_(n=0) ^∞ (1/(n!)) =e Σ_(n=0) ^∞ (n^2 /(n!)) =Σ_(n=1) ^∞ (n/((n−1)!)) = Σ_(n=0) ^∞ ((n+1)/(n!)) = Σ_(n=1) ^∞ (1/((n−1)!)) +Σ_(n=0) ^∞ (1/(n!)) = Σ_(n=0) ^∞ (1/(n!)) +e =2e ⇒ Σ_(n=0) ^∞ ((n^2 −2)/(n!)) =2e −2e =0 .

$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{n}^{\mathrm{2}} −\mathrm{2}}{{n}!}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{{n}^{\mathrm{2}} }{{n}!}\:−\mathrm{2}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\:{but}\:{we}\:{have} \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\:={e} \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{n}^{\mathrm{2}} }{{n}!}\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{n}}{\left({n}−\mathrm{1}\right)!}\:=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{n}+\mathrm{1}}{{n}!} \\ $$$$=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)!}\:\:+\sum_{{n}=\mathrm{0}} ^{\infty} \:\:\frac{\mathrm{1}}{{n}!} \\ $$$$=\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{\mathrm{1}}{{n}!}\:+{e}\:=\mathrm{2}{e}\:\Rightarrow \\ $$$$\sum_{{n}=\mathrm{0}} ^{\infty} \:\frac{{n}^{\mathrm{2}} −\mathrm{2}}{{n}!}\:=\mathrm{2}{e}\:−\mathrm{2}{e}\:=\mathrm{0}\:. \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com