Question
$$\frac{1}{2}{x^2} + \frac{2}{3}{x^3} + \frac{3}{4}{x^4} + \frac{4}{5}{x^5} + .....{\text{is}}$$
A.
$$\frac{x}{{1 + x}} + \log \left( {1 + x} \right)$$
B.
$$\frac{x}{{1 - x}} + \log \left( {1 + x} \right)$$
C.
$$ - \frac{x}{{1 + x}} + \log \left( {1 + x} \right)$$
D.
$$\frac{x}{{1 - x}} + \log \left( {1 - x} \right)$$
Answer :
$$\frac{x}{{1 - x}} + \log \left( {1 - x} \right)$$
Solution :
$$\eqalign{
& \frac{1}{2}{x^2} + \frac{2}{3}{x^3} + \frac{3}{4}{x^4} + \frac{4}{5}{x^5} + ..... \cr
& = \left( {1 - \frac{1}{2}} \right){x^2} + \left( {1 - \frac{1}{3}} \right){x^3} + \left( {1 - \frac{1}{4}} \right){x^4} + \left( {1 - \frac{1}{5}} \right){x^5} + ..... \cr
& = \left( {{x^2} + {x^3} + {x^4} + {x^5} + .....} \right) + \left( { - \frac{{{x^2}}}{2} - \frac{{{x^3}}}{3} - \frac{{{x^4}}}{4} - \frac{{{x^5}}}{5}.....} \right) \cr
& = \left( {x + {x^2} + {x^3} + .....} \right) + \left( { - \frac{{{x^2}}}{2} - \frac{{{x^3}}}{3} - \frac{{{x^4}}}{4} - \frac{{{x^5}}}{5}.....} \right) \cr
& = \frac{x}{{1 - x}} + \log \left( {1 - x} \right) \cr} $$