Question

Find the sum of n terms of the series $\frac{2}{1.5}-\frac{4}{5.7}+\frac{8}{7.17}-\frac{16}{17.31}+\frac{32}{31.65}-....$

Answer 1

$\Rightarrow$ When n is odd

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

And When n is even

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

$3S_{2m}=(\frac{1}{1}+\frac{1}{5})-(\frac{1}{5}+\frac{1}{7})+(\frac{1}{7}+\frac{1}{17})-.....-(\frac{1}{2^{2m+1}+1}+\frac{1}{2^{2m+2}-1})$

and $3S_{2m+1}=(\frac{1}{1}+\frac{1}{5})-(\frac{1}{5}+\frac{1}{7})+.....+(\frac{1}{2^{2m+1}-1}+\frac{1}{2^{2m+2}+1})$

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

$3S_{2m+1}=1+\frac{1}{2^{2m+1}+1}$

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