The correct matching of List-I from List-II i | vedclass

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(C) $(1-x)^{-n} = 1+nx+\frac{n(n+1)}{2!}x^2+\dots$ for $|x| < 1$. This matches $(iii)$.
$(B)$ $(1+x)^{-n} = 1-nx+\frac{n(n+1)}{2!}x^2-\dots$ for $

Vedclass · Accepted Answer

$(A)-(iii), (B)-(ii), (C)-(iv), (D)-(v)$

Vedclass · Answer

(C) $(1-x)^{-n} = 1+nx+\frac{n(n+1)}{2!}x^2+\dots$ for $|x| < 1$. This matches $(iii)$.
$(B)$ $(1+x)^{-n} = 1-nx+\frac{n(n+1)}{2!}x^2-\dots$ for $|x| < 1$. This matches $(ii)$.
$(C)$ For $x>1$,the series $1+\frac{1}{x}+\frac{1}{x^2}+\dots$ is a geometric progression with first term $a=1$ and common ratio $r=\frac{1}{x}$. The sum is $S = \frac{a}{1-r} = \frac{1}{1-\frac{1}{x}} = \frac{x}{x-1}$. This matches $(iv)$.
$(D)$ Let $S = 1-\frac{2}{x^2}+\frac{3}{x^4}-\frac{4}{x^6}+\dots$. This is of the form $(1+y)^{-2}$ where $y = \frac{1}{x^2}$.
$(1+y)^{-2} = 1-2y+3y^2-4y^3+\dots = 1-\frac{2}{x^2}+\frac{3}{x^4}-\frac{4}{x^6}+\dots$.
Thus,$S = (1+\frac{1}{x^2})^{-2} = (\frac{x^2+1}{x^2})^{-2} = \frac{x^4}{(x^2+1)^2}$. This matches $(v)$.
Therefore,the correct matching is $(A)-(iii), (B)-(ii), (C)-(iv), (D)-(v)$.

List-$I$	List-$II$
$(A)$ $(1-x)^{-n}$	$(i)$ $\frac{x}{x+1}$
$(B)$ $(1+x)^{-n}$	$(ii)$ $1-nx+\frac{n(n+1)}{2!}x^2-\dots$ if $\|x\| < 1$
$(C)$ If $x>1$,then $1+\frac{1}{x}+\frac{1}{x^2}+\dots$ is	$(iii)$ $1+nx+\frac{n(n+1)}{2!}x^2+\dots$ if $\|x\| < 1$
$(D)$ If $\|x\|>1$,then $1-\frac{2}{x^2}+\frac{3}{x^4}-\frac{4}{x^6}+\dots$ is	$(iv)$ $\frac{x}{x-1}$
	$(v)$ $\frac{x^4}{(x^2+1)^2}$
	$(vi)$ $\frac{x^4}{(x^2-1)^2}$

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