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  4. For x>0, the sum of the series (1/1+x)-((1-x)/(1+x)2) +((1-x)2/(1+x)3)-⋅s ∞ is equal to

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Q. For $x>0,$ the sum of the series $\frac{1}{1+x}-\frac{(1-x)}{(1+x)^{2}}$ $+\frac{(1-x)^{2}}{(1+x)^{3}}-\cdots \infty$ is equal to

Sequences and Series

Solution:

$S=\frac{1}{1+x}-\frac{(1-x)}{(1+x)^{2}}+\frac{(1-x)^{2}}{(1+x)^{3}}-\cdots \infty$
The above series is an infinite $G.P$.
whose first term $=\frac{1}{1+x}$ and common ratio $=\frac{-(1-x)}{(1+x)}$
$\therefore \,\, S_{\infty}=\frac{\frac{1}{1+x}}{1+\left(\frac{1-x}{1+x}\right)}=\frac{1}{2}$