Question Number 146722 by SOMEDAVONG last updated on 15/Jul/21
![D=lim_(n→+∝) (n^2 /(n−1))[((sin((88)/n))/(1+2)) + ((sin((88)/n))/(1+2+3)) + ...+ ((sin((88)/n))/(1+2+3+...+n))]](https://www.tinkutara.com/question/Q146722.png)
$$\mathrm{D}=\underset{\mathrm{n}\rightarrow+\propto} {\mathrm{lim}}\frac{\mathrm{n}^{\mathrm{2}} }{\mathrm{n}−\mathrm{1}}\left[\frac{\mathrm{sin}\frac{\mathrm{88}}{\mathrm{n}}}{\mathrm{1}+\mathrm{2}}\:+\:\frac{\mathrm{sin}\frac{\mathrm{88}}{\mathrm{n}}}{\mathrm{1}+\mathrm{2}+\mathrm{3}}\:+\:…+\:\frac{\mathrm{sin}\frac{\mathrm{88}}{\mathrm{n}}}{\mathrm{1}+\mathrm{2}+\mathrm{3}+…+\mathrm{n}}\right] \\ $$
Answered by Olaf_Thorendsen last updated on 15/Jul/21
![S_n = (n^2 /(n−1))[Σ_(k=2) ^n ((sin((88)/n))/(Σ_(p=1) ^k p))] S_n = (n^2 /(n−1))[Σ_(k=2) ^n ((sin((88)/n))/((k(k+1))/2))] S_n = (n^2 /(n−1))sin((88)/n)(Σ_(k=2) ^n (2/(k(k+1)))) S_n = ((2n^2 )/(n−1))sin((88)/n)(Σ_(k=2) ^n (1/k)−(1/(k+1))) S_n = ((2n^2 )/(n−1))sin((88)/n)((1/2)−(1/(n+1))) S_n = ((2n^2 )/(n−1))sin((88)/n)(((n−1)/(2(n+1)))) S_n = (n^2 /(n+1))sin((88)/n) S_n ∼_∞ (n^2 /(n+1))×((88)/n) = ((88n)/(n+1)) →_∞ 88](https://www.tinkutara.com/question/Q146726.png)
$$\mathrm{S}_{{n}} \:=\:\frac{{n}^{\mathrm{2}} }{{n}−\mathrm{1}}\left[\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{\mathrm{sin}\frac{\mathrm{88}}{{n}}}{\underset{{p}=\mathrm{1}} {\overset{{k}} {\sum}}{p}}\right] \\ $$$$\mathrm{S}_{{n}} \:=\:\frac{{n}^{\mathrm{2}} }{{n}−\mathrm{1}}\left[\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{\mathrm{sin}\frac{\mathrm{88}}{{n}}}{\frac{{k}\left({k}+\mathrm{1}\right)}{\mathrm{2}}}\right] \\ $$$$\mathrm{S}_{{n}} \:=\:\frac{{n}^{\mathrm{2}} }{{n}−\mathrm{1}}\mathrm{sin}\frac{\mathrm{88}}{{n}}\left(\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{\mathrm{2}}{{k}\left({k}+\mathrm{1}\right)}\right) \\ $$$$\mathrm{S}_{{n}} \:=\:\frac{\mathrm{2}{n}^{\mathrm{2}} }{{n}−\mathrm{1}}\mathrm{sin}\frac{\mathrm{88}}{{n}}\left(\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{k}}−\frac{\mathrm{1}}{{k}+\mathrm{1}}\right) \\ $$$$\mathrm{S}_{{n}} \:=\:\frac{\mathrm{2}{n}^{\mathrm{2}} }{{n}−\mathrm{1}}\mathrm{sin}\frac{\mathrm{88}}{{n}}\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\right) \\ $$$$\mathrm{S}_{{n}} \:=\:\frac{\mathrm{2}{n}^{\mathrm{2}} }{{n}−\mathrm{1}}\mathrm{sin}\frac{\mathrm{88}}{{n}}\left(\frac{{n}−\mathrm{1}}{\mathrm{2}\left({n}+\mathrm{1}\right)}\right) \\ $$$$\mathrm{S}_{{n}} \:=\:\frac{{n}^{\mathrm{2}} }{{n}+\mathrm{1}}\mathrm{sin}\frac{\mathrm{88}}{{n}} \\ $$$$\mathrm{S}_{{n}} \:\underset{\infty} {\sim}\:\frac{{n}^{\mathrm{2}} }{{n}+\mathrm{1}}×\frac{\mathrm{88}}{{n}}\:=\:\frac{\mathrm{88}{n}}{{n}+\mathrm{1}}\:\underset{\infty} {\rightarrow}\:\mathrm{88} \\ $$