Question

Find the sum of n terms of the series $\frac{5}{1.2}\cdot \frac{1}{3}+\frac{7}{2.3}\cdot \frac{1}{3^2}+\frac{9}{3.4}\cdot \frac{1}{3^3}+\frac{11}{4.5}\cdot \frac{1}{3^4}+....$

Answer 1

Here $u_n=\frac{2n+3}{n(n+1)}\cdot \frac{1}{3^n}$
Assuming $\frac{2n+3}{n(n+1)}=\frac{A}{n}+\frac{B}{n+1}$,
We find $A=3,B=-1$.
Hence $u_n=\left(\frac{3}{n}-\frac{1}{n+1}\right)\frac{1}{3^n}=\frac{1}{n}\cdot \frac{1}{3^{n-1}}-\frac{1}{n+1}\cdot \frac{1}{3^n}$,
and therefore $S_n=1-\frac{1}{n+1}\cdot \frac{1}{3^n}$.