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Question Number 41236 by maxmathsup by imad last updated on 04/Aug/18

find the value of  Σ_(n=2) ^∞    ((3n^2  +1)/((n^2 −1)^3 ))

$${find}\:{the}\:{value}\:{of}\:\:\sum_{{n}=\mathrm{2}} ^{\infty} \:\:\:\frac{\mathrm{3}{n}^{\mathrm{2}} \:+\mathrm{1}}{\left({n}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{3}} } \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 04/Aug/18

(n+1)^3 −(n−1)^3 =(n^3 +3n^2 +3n+1)−(n^3 −3n^2 +3n−1)  =6^ n^2 +2  Σ_(n=2) ^∞ (1/2).(((n+1)^3 −(n−1)^3 )/((n+1)^3 (n−1)^3 ))  =(1/2)Σ_(n=2) ^∞ (1/((n−1)^3 ))−(1/((n+1)^3 ))=(1/2)Σ_2 ^∞ T_n   T_n =[(1/((n−1)^3 )) −(1/((n+1)^3 ))]  T_2 =(1/1^3 )−(1/3^3 )  T_3 =(1/2^3 )−(1/4^3 )  T_4 =(1/3^3 )−(1/5^3 )  T_5 =(1/4^3 )−(1/6^3 )  ....  ....  T_n =(1/((n−1)^3 ))−(1/((n+1)^3 ))  s=(1/1^3 )+(1/2^3 )−(1/((n+1)^3 ))  (1/2)Σ_2 ^∞ T_n  =(1/2)(1+(1/8))−lim_(n→∞) (1/((n+1)^3 ))=(9/(16))−0=(9/(16))    ]

$$\left({n}+\mathrm{1}\right)^{\mathrm{3}} −\left({n}−\mathrm{1}\right)^{\mathrm{3}} =\left({n}^{\mathrm{3}} +\mathrm{3}{n}^{\mathrm{2}} +\mathrm{3}{n}+\mathrm{1}\right)−\left({n}^{\mathrm{3}} −\mathrm{3}{n}^{\mathrm{2}} +\mathrm{3}{n}−\mathrm{1}\right) \\ $$$$=\mathrm{6}^{} {n}^{\mathrm{2}} +\mathrm{2} \\ $$$$\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\mathrm{2}}.\frac{\left({n}+\mathrm{1}\right)^{\mathrm{3}} −\left({n}−\mathrm{1}\right)^{\mathrm{3}} }{\left({n}+\mathrm{1}\right)^{\mathrm{3}} \left({n}−\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\underset{{n}=\mathrm{2}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)^{\mathrm{3}} }−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{3}} }=\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{2}} {\overset{\infty} {\sum}}{T}_{{n}} \\ $$$${T}_{{n}} =\left[\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)^{\mathrm{3}} }\:−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{3}} }\right] \\ $$$${T}_{\mathrm{2}} =\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{3}} } \\ $$$${T}_{\mathrm{3}} =\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{3}} } \\ $$$${T}_{\mathrm{4}} =\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{5}^{\mathrm{3}} } \\ $$$${T}_{\mathrm{5}} =\frac{\mathrm{1}}{\mathrm{4}^{\mathrm{3}} }−\frac{\mathrm{1}}{\mathrm{6}^{\mathrm{3}} } \\ $$$$.... \\ $$$$.... \\ $$$${T}_{{n}} =\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)^{\mathrm{3}} }−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$${s}=\frac{\mathrm{1}}{\mathrm{1}^{\mathrm{3}} }+\frac{\mathrm{1}}{\mathrm{2}^{\mathrm{3}} }−\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\underset{\mathrm{2}} {\overset{\infty} {\sum}}{T}_{{n}} \:=\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{8}}\right)−\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{3}} }=\frac{\mathrm{9}}{\mathrm{16}}−\mathrm{0}=\frac{\mathrm{9}}{\mathrm{16}} \\ $$$$ \\ $$$$\left.\right] \\ $$

Commented by prof Abdo imad last updated on 04/Aug/18

your answer is correct sir Tanmay thanks

$${your}\:{answer}\:{is}\:{correct}\:{sir}\:{Tanmay}\:{thanks} \\ $$

Commented by tanmay.chaudhury50@gmail.com last updated on 04/Aug/18

thank you sir...

$${thank}\:{you}\:{sir}... \\ $$

Answered by prof Abdo imad last updated on 04/Aug/18

S =lim_(n→+∞)  S_N   with S_N =Σ_(n=2) ^N  ((3n^(2 ) +1)/((n−1)^3 (n+1)^3 ))  but we have (n+1)^3 −(n−1)^3   =2((n+1)^2  +(n+1)(n−1) +(n−1)^3 )  =2(n^2  +2n+1 +n^2 −1 +n^2 −2n+1)  =2(3n^2 +1)⇒S_N =(1/2)Σ_(n=2) ^N (((n+1)^3 −(n−1)^3 )/((n−1)^3 (n+1)^3 ))  =(1/2){Σ_(n=2) ^N  (1/((n−1)^3 )) −Σ_(n=2) ^N  (1/((n+1)^3 ))}  but  Σ_(n=2) ^N  (1/((n−1)^3 )) =Σ_(n=1) ^(N−1)  (1/n^3 ) =ξ_(N−1) (3)  Σ_(n=2) ^N  (1/((n+1)^3 )) =Σ_(n=3) ^(N+1)  (1/n^3 ) =ξ_(N+1) (3)−1−(1/8) ⇒  S_N =(1/2){ ξ_(N−1) (3)−ξ_(N+1) (3) +(9/8)}but  lim_(N→+∞) ξ_(N−1) (3)−ξ_(N+1) (3)=ξ(3)−ξ(3)=0⇒  lim_(N→+∞)  S_N = (9/(16)) =S

$${S}\:={lim}_{{n}\rightarrow+\infty} \:{S}_{{N}} \:\:{with}\:{S}_{{N}} =\sum_{{n}=\mathrm{2}} ^{{N}} \:\frac{\mathrm{3}{n}^{\mathrm{2}\:} +\mathrm{1}}{\left({n}−\mathrm{1}\right)^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$${but}\:{we}\:{have}\:\left({n}+\mathrm{1}\right)^{\mathrm{3}} −\left({n}−\mathrm{1}\right)^{\mathrm{3}} \\ $$$$=\mathrm{2}\left(\left({n}+\mathrm{1}\right)^{\mathrm{2}} \:+\left({n}+\mathrm{1}\right)\left({n}−\mathrm{1}\right)\:+\left({n}−\mathrm{1}\right)^{\mathrm{3}} \right) \\ $$$$=\mathrm{2}\left({n}^{\mathrm{2}} \:+\mathrm{2}{n}+\mathrm{1}\:+{n}^{\mathrm{2}} −\mathrm{1}\:+{n}^{\mathrm{2}} −\mathrm{2}{n}+\mathrm{1}\right) \\ $$$$=\mathrm{2}\left(\mathrm{3}{n}^{\mathrm{2}} +\mathrm{1}\right)\Rightarrow{S}_{{N}} =\frac{\mathrm{1}}{\mathrm{2}}\sum_{{n}=\mathrm{2}} ^{{N}} \frac{\left({n}+\mathrm{1}\right)^{\mathrm{3}} −\left({n}−\mathrm{1}\right)^{\mathrm{3}} }{\left({n}−\mathrm{1}\right)^{\mathrm{3}} \left({n}+\mathrm{1}\right)^{\mathrm{3}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\sum_{{n}=\mathrm{2}} ^{{N}} \:\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)^{\mathrm{3}} }\:−\sum_{{n}=\mathrm{2}} ^{{N}} \:\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{3}} }\right\}\:\:{but} \\ $$$$\sum_{{n}=\mathrm{2}} ^{{N}} \:\frac{\mathrm{1}}{\left({n}−\mathrm{1}\right)^{\mathrm{3}} }\:=\sum_{{n}=\mathrm{1}} ^{{N}−\mathrm{1}} \:\frac{\mathrm{1}}{{n}^{\mathrm{3}} }\:=\xi_{{N}−\mathrm{1}} \left(\mathrm{3}\right) \\ $$$$\sum_{{n}=\mathrm{2}} ^{{N}} \:\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{3}} }\:=\sum_{{n}=\mathrm{3}} ^{{N}+\mathrm{1}} \:\frac{\mathrm{1}}{{n}^{\mathrm{3}} }\:=\xi_{{N}+\mathrm{1}} \left(\mathrm{3}\right)−\mathrm{1}−\frac{\mathrm{1}}{\mathrm{8}}\:\Rightarrow \\ $$$${S}_{{N}} =\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\xi_{{N}−\mathrm{1}} \left(\mathrm{3}\right)−\xi_{{N}+\mathrm{1}} \left(\mathrm{3}\right)\:+\frac{\mathrm{9}}{\mathrm{8}}\right\}{but} \\ $$$${lim}_{{N}\rightarrow+\infty} \xi_{{N}−\mathrm{1}} \left(\mathrm{3}\right)−\xi_{{N}+\mathrm{1}} \left(\mathrm{3}\right)=\xi\left(\mathrm{3}\right)−\xi\left(\mathrm{3}\right)=\mathrm{0}\Rightarrow \\ $$$${lim}_{{N}\rightarrow+\infty} \:{S}_{{N}} =\:\frac{\mathrm{9}}{\mathrm{16}}\:={S} \\ $$

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