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let-S-n-k-2-n-1-k-k-2-1-1-calculate-S-n-interms-of-H-n-H-n-k-1-n-1-k-2-find-lim-n-S-n-

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Question Number 40046 by abdo mathsup 649 cc last updated on 15/Jul/18
let S_n = Σ_(k=2) ^n (((−1)^k )/(k^2 −1)) 1) calculate S_n interms of H_n ( H_n =Σ_(k=1) ^n (1/k)) 2) find lim_(n→+∞) S_n
$${let}\:{S}_{{n}} =\:\sum_{{k}=\mathrm{2}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}^{\mathrm{2}} −\mathrm{1}} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:\:{S}_{{n}} \:\:{interms}\:{of}\:{H}_{{n}} \\ $$$$\left(\:{H}_{{n}} =\sum_{{k}=\mathrm{1}} ^{{n}} \:\frac{\mathrm{1}}{{k}}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{n}\rightarrow+\infty} \:{S}_{{n}} \\ $$$$ \\ $$
Commented by math khazana by abdo last updated on 26/Jul/18
1) we have S_n = Σ_(k=2) ^n (((−1)^k )/(k^2 −1)) =(1/2)Σ_(k=2) ^n (−1)^k { (1/(k−1)) −(1/(k+1))} =(1/2){ Σ_(k=2) ^n (((−1)^k )/(k−1)) −Σ_(k=2) ^n (((−1)^k )/(k+1))} but Σ_(k=2) ^n (((−1)^k )/(k−1)) =Σ_(k=1) ^(n−1) (((−1)^(k+1) )/k) =Σ_(p=1) ^([((n−1)/2)]) ((−1)/(2p)) + Σ_(p=0) ^([((n−2)/2)]) (1/(2p+1)) =−(1/2)H_([((n−1)/2)]) +Σ_(p=0) ^([((n−2)/2)]) (1/(2p+1)) Σ_(k=2) ^n (((−1)^k )/(k+1)) = Σ_(k=3) ^(n+1) (((−1)^(k−1) )/k) =Σ_(k=1) ^(n+1) (((−1)^(k−1) )/k) −(1−(1/2))=Σ_(k=1) ^(n+1) (((−1)^(k−1) )/k) −(1/2) =Σ_(p=1) ^([((n+1)/2)]) ((−1)/(2p)) +Σ_(p=0) ^([(n/2)]) (1/(2p+1)) −(1/2) ⇒ S_n = (1/2){−(1/2)H_([((n−1)/2)]) +Σ_(p=0) ^([((n−2)/2)]) (1/(2p+1)) +(1/2)H_([((n+1)/2)]) −Σ_(p=0) ^([(n/2)]) (1/(2p+1)) +(1/2)}
$$\left.\mathrm{1}\right)\:{we}\:{have}\:{S}_{{n}} =\:\sum_{{k}=\mathrm{2}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}^{\mathrm{2}} −\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\sum_{{k}=\mathrm{2}} ^{{n}} \:\left(−\mathrm{1}\right)^{{k}} \left\{\:\frac{\mathrm{1}}{{k}−\mathrm{1}}\:−\frac{\mathrm{1}}{{k}+\mathrm{1}}\right\} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\sum_{{k}=\mathrm{2}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}−\mathrm{1}}\:−\sum_{{k}=\mathrm{2}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}+\mathrm{1}}\right\}\:{but} \\ $$$$\sum_{{k}=\mathrm{2}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}−\mathrm{1}}\:=\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} }{{k}} \\ $$$$=\sum_{{p}=\mathrm{1}} ^{\left[\frac{{n}−\mathrm{1}}{\mathrm{2}}\right]} \frac{−\mathrm{1}}{\mathrm{2}{p}}\:\:+\:\sum_{{p}=\mathrm{0}} ^{\left[\frac{{n}−\mathrm{2}}{\mathrm{2}}\right]} \:\:\frac{\mathrm{1}}{\mathrm{2}{p}+\mathrm{1}} \\ $$$$=−\frac{\mathrm{1}}{\mathrm{2}}{H}_{\left[\frac{{n}−\mathrm{1}}{\mathrm{2}}\right]} \:\:\:+\sum_{{p}=\mathrm{0}} ^{\left[\frac{{n}−\mathrm{2}}{\mathrm{2}}\right]} \:\:\frac{\mathrm{1}}{\mathrm{2}{p}+\mathrm{1}} \\ $$$$\sum_{{k}=\mathrm{2}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}+\mathrm{1}}\:=\:\sum_{{k}=\mathrm{3}} ^{{n}+\mathrm{1}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}} \\ $$$$=\sum_{{k}=\mathrm{1}} ^{{n}+\mathrm{1}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}}\:−\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{2}}\right)=\sum_{{k}=\mathrm{1}} ^{{n}+\mathrm{1}} \:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}}\:−\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$=\sum_{{p}=\mathrm{1}} ^{\left[\frac{{n}+\mathrm{1}}{\mathrm{2}}\right]} \frac{−\mathrm{1}}{\mathrm{2}{p}}\:\:+\sum_{{p}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{2}}\right]} \:\:\:\frac{\mathrm{1}}{\mathrm{2}{p}+\mathrm{1}}\:−\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow \\ $$$${S}_{{n}} =\:\frac{\mathrm{1}}{\mathrm{2}}\left\{−\frac{\mathrm{1}}{\mathrm{2}}{H}_{\left[\frac{{n}−\mathrm{1}}{\mathrm{2}}\right]} \:\:\:+\sum_{{p}=\mathrm{0}} ^{\left[\frac{{n}−\mathrm{2}}{\mathrm{2}}\right]} \:\:\frac{\mathrm{1}}{\mathrm{2}{p}+\mathrm{1}}\:+\frac{\mathrm{1}}{\mathrm{2}}{H}_{\left[\frac{{n}+\mathrm{1}}{\mathrm{2}}\right]} \right. \\ $$$$\left.−\sum_{{p}=\mathrm{0}} ^{\left[\frac{{n}}{\mathrm{2}}\right]} \:\:\:\frac{\mathrm{1}}{\mathrm{2}{p}+\mathrm{1}}\:+\frac{\mathrm{1}}{\mathrm{2}}\right\} \\ $$$$ \\ $$
Commented by math khazana by abdo last updated on 26/Jul/18
2) we have S_n =(1/2){ Σ_(k=2) ^n (((−1)^k )/(k−1)) −Σ_(k=2) ^n (((−1)^k )/(k+1))} but Σ_(k=2) ^n (((−1)^k )/(k−1)) =Σ_(k=1) ^(n−1) (((−1)^(k+1) )/k) =−Σ_(k=1) ^(n−1) (((−1)^k )/k) Σ_(k=2) ^n (((−1)^k )/(k+1)) = Σ_(k=3) ^(n+1) (((−1)^(k−1) )/k) =−Σ_(k=3) ^(n+1) (((−1)^k )/k) =−{ Σ_(k=1) ^(n−1) (((−1)^k )/k) +(((−1)^n )/n) +(((−1)^(n+1) )/(n+1)) −(((−1))/1) −(1/2)} = −Σ_(k=1) ^(n−1) (((−1)^k )/k) −(((−1)^n )/n) +(((−1)^n )/(n+1)) −(1/2) ⇒ S_n =(1/2){−Σ_(k=1) ^(n−1) (((−1)^k )/k) +Σ_(k=1) ^(n−1) (((−1)^k )/k) +(((−1)^n )/n) −(((−1)^n )/(n+1)) +(1/2)} S_n = (1/2){ (((−1)^n )/n) −(((−1)^n )/(n+1)) +(1/2)} ⇒ lim_(n→+∞) S_n =(1/4) .
$$\left.\mathrm{2}\right)\:{we}\:{have}\:\:{S}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\sum_{{k}=\mathrm{2}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}−\mathrm{1}}\:−\sum_{{k}=\mathrm{2}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}+\mathrm{1}}\right\} \\ $$$${but}\:\sum_{{k}=\mathrm{2}} ^{{n}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}−\mathrm{1}}\:=\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}+\mathrm{1}} }{{k}}\:=−\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}} \\ $$$$\sum_{{k}=\mathrm{2}} ^{{n}} \:\:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}+\mathrm{1}}\:=\:\sum_{{k}=\mathrm{3}} ^{{n}+\mathrm{1}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} }{{k}} \\ $$$$=−\sum_{{k}=\mathrm{3}} ^{{n}+\mathrm{1}} \:\:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}}\:=−\left\{\:\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}}\:+\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}}\:+\frac{\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}\right. \\ $$$$\left.−\frac{\left(−\mathrm{1}\right)}{\mathrm{1}}\:−\frac{\mathrm{1}}{\mathrm{2}}\right\} \\ $$$$=\:−\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}}\:−\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}}\:+\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+\mathrm{1}}\:−\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow \\ $$$${S}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}}\left\{−\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}}\:+\sum_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{{k}}\:+\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}}\right. \\ $$$$\left.−\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+\mathrm{1}}\:+\frac{\mathrm{1}}{\mathrm{2}}\right\} \\ $$$${S}_{{n}} =\:\frac{\mathrm{1}}{\mathrm{2}}\left\{\:\:\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}}\:−\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}+\mathrm{1}}\:+\frac{\mathrm{1}}{\mathrm{2}}\right\}\:\Rightarrow \\ $$$${lim}_{{n}\rightarrow+\infty} \:\:{S}_{{n}} =\frac{\mathrm{1}}{\mathrm{4}}\:. \\ $$

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