For x0, the sum of the series (1/1+x)-((1-x)/(1+x)2) +((1-x)2/(1+x)3)-⋅s ∞ is equal to

Question

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 (A) (1/4) (B) (1/2) (C) (3/4) (D) 1

Tardigrade Admin · Accepted Answer

S=(1/1+x)-((1-x)/(1+x)2)+((1-x)2/(1+x)3)-⋅s ∞ The above series is an infinite G.P. whose first term =(1/1+x) and common ratio =(-(1-x)/(1+x)) ∴ S∞=((1/1+x)/1+((1-x)1+x))=(1/2)

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

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{3}{4}$

1

Q. For x>0, the sum of the series 1+x1​−(1+x)2(1−x)​ +(1+x)3(1−x)2​−⋯∞ is equal to

41​

21​

43​

1

S=1+x1​−(1+x)2(1−x)​+(1+x)3(1−x)2​−⋯∞ The above series is an infinite G.P. whose first term =1+x1​ and common ratio =(1+x)−(1−x)​ ∴S∞​=1+(1+x1−x​)1+x1​​=21​

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

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{3}{4}$