NIMCET 2023 — Mathematics PYQ

$${\log }_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\frac{x^5}{5}-\cdots -\frac{x^{100}}{100}+\cdots$$
$$<br>\ln (1+x^2)=x^2-\frac{x^4}{2}+\frac{x^6}{3}-\frac{x^8}{4}+\cdots -\frac{(x^2{)}^{50}}{50}$$
Coefficient of $x^{100}$ in $\ln \left(\frac{1+x}{1+x^2}\right)$
Coefficient of $x^{100}$ in $\ln (1+x)-\ln (1+x^2)$
$$=-\frac{1}{100}+\frac{1}{50}=0.01$$