Teaching MPPSC Assistant Professor Mock Test Series 2025 Mathematical Science Analysis Sequences & Series (Convergence)
For −1 ≤ x ≤ 1, if f(x) is the sum of the convergent power series \(x+\frac{x^{2}}{2^{2}}+\frac{x^{3}}{3^{2}}+\cdots+\frac{x^{n}}{n^{2}}+\cdots\) then \(f\left(\frac{1}{2}\right)\) is equal to
1
\(\int_{0}^{\frac{1}{2}} \frac{\ln (1-t)}{t} d t\)
2
\(-\int_{0}^{\frac{1}{2}} \frac{\ln (1-t)}{t} d t\)
3
\(\int_{0}^{\frac{1}{2}} \mathrm{t} \ln (1+t) d t\)
4
\(\int_{0}^{\frac{1}{2}} \mathrm{t} \ln (1-t) d t\)