Question
If $$\left| x \right| < \frac{1}{2},$$ what is the value of $$1 + n\left[ {\frac{x}{{1 - x}}} \right] + \left[ {\frac{{n\left( {n + 1} \right)}}{{2\,!}}} \right]{\left[ {\frac{x}{{1 - x}}} \right]^2} + .....\,\infty \,?$$
A.
$${\left[ {\frac{{1 - x}}{{1 - 2x}}} \right]^n}$$
B.
$${\left( {1 - x} \right)^n}$$
C.
$${\left[ {\frac{{1 - 2x}}{{1 - x}}} \right]^n}$$
D.
$${\left( {\frac{1}{{1 - x}}} \right)^n}$$
Answer :
$${\left[ {\frac{{1 - x}}{{1 - 2x}}} \right]^n}$$
Solution :
Given that $$1 + n\left[ {\frac{x}{{1 - x}}} \right] + \left[ {\frac{{n\left( {n + 1} \right)}}{{2\,!}}} \right]{\left[ {\frac{x}{{1 - x}}} \right]^2} + .....\,\infty\,\, $$ is expansion of $${\left[ {1 - \frac{x}{{1 - x}}} \right]^{ - n}}.$$
So, it is $$ = {\left[ {1 - \frac{x}{{1 - x}}} \right]^{ - n}}$$
$$ = {\left[ {\frac{{1 - x - x}}{{1 - x}}} \right]^{ - n}} = {\left[ {\frac{{1 - x}}{{1 - 2x}}} \right]^n}$$