Question

Find the sum of n terms of the series $\frac{5}{1.2}-\frac{3}{2.3}+\frac{9}{3.4}-\frac{7}{4.5}+\frac{13}{5.6}-\frac{11}{6.7}+\frac{17}{7.8}-.....$

Answer 1

For the odd term

$\Rightarrow u_n=\frac{2n+3}{n(n+1)}=\frac{3}{n}-\frac{1}{n+1}$

And For the even term,

$u_n=\frac{2n-1}{n(n+1)}=\frac{3}{n+1}-\frac{1}{n}$

$S_{even}=(\frac{3}{1}-\frac{1}{2})+(\frac{1}{2}-\frac{3}{3})+(\frac{3}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{3}{5})+....+(\frac{1}{2m}-\frac{1}{2m+1})$

$=3-\frac{1}{2m+1}=\frac{6m}{2m+1}$

$S_{odd}=(\frac{3}{1}-\frac{1}{2})+(\frac{1}{2}-\frac{3}{3})+(\frac{3}{3}-\frac{1}{4})+(\frac{1}{4}-\frac{3}{5})+....+(\frac{3}{2m+1}-\frac{1}{2m+2})$

$=3-\frac{1}{2m+2}=\frac{6m+5}{2m+2}$

$S_n=3-\frac{2+{(-1)}^n}{n+1}$