Question:
If $0 to :
Correct Option: 1
Solution:
Let $\mathrm{t}=\frac{3}{2} \mathrm{x}^{2}+\frac{5}{3} \mathrm{x}^{3}+\frac{7}{4} \mathrm{x}^{4}+\ldots . \infty$
$=\left(2-\frac{1}{2}\right) x^{2}+\left(2-\frac{1}{3}\right) x^{3}+\left(2-\frac{1}{4}\right) x^{4}$
$+\ldots . \infty$
$=2\left(x^{2}+x^{3}+x^{4}+\ldots \infty\right)-\left(\frac{x^{2}}{2}+\frac{x^{3}}{3}+\frac{x^{4}}{4}+\ldots \infty\right)$
$=\frac{2 x^{2}}{1-x}-(\ell n(1-x)-x)$
$\Rightarrow t=\frac{2 x^{2}}{1-x}+x-\ell n(1-x)$
$\Rightarrow \mathrm{t}=\frac{\mathrm{x}(1+\mathrm{x})}{1-\mathrm{x}}-\ell \mathrm{n}(1-\mathrm{x})$