A) \[\frac{2}{1-{{x}^{2}}}\]
B) \[\frac{3}{{{(1-x)}^{2}}}\]
C) \[\frac{x}{1-{{x}^{2}}}\]
D) \[\frac{x}{{{(1-x)}^{2}}}\]
Correct Answer: D
Solution :
(d): Let \[S=1+2x+3{{x}^{2}}+4{{x}^{3}}+\] ?. (1) \[xS=1x+2{{x}^{2}}+3{{x}^{3}}+\] ?. (2) Eq. (1) ? Eq. (2) gives \[S(1-x)=1+x+{{x}^{2}}+{{x}^{3}}+{{x}^{4}}+\]??. Now, \[1+x+{{x}^{2}}+\]??. is an infinite GP with a = 1, r = x and |r| = |x|<1 \[\therefore \] Sum of the series \[=\frac{1}{1-x}\] \[\therefore \]\[S(1-x)=\frac{1}{(1-x)}\] \[\therefore \]\[S=\frac{1}{{{(1-x)}^{2}}}\]
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