Question

For \(-1\le x\le1\), if f (x) is the sum of the convergent power series \(x+\frac{x^2}{2^2}+\frac{x^3}{3^2}+...+\frac{x^n}{n^2}+...\) then \(f(\frac{1}{2})\) is equal to

• A. \(\displaystyle\int\limits^{\frac{1}{2}}_{0}\frac{In(1-t)}{t}dt.\)
• B. \(-\displaystyle\int\limits^{\frac{1}{2}}_{0}\frac{In(1-t)}{t}dt.\)
• C. \(\displaystyle\int\limits^{\frac{1}{2}}_{0}tIn(1+t)dt.\)
• D. \(\displaystyle\int\limits^{\frac{1}{2}}_{0}tIn(1-t)dt.\)

Answer 1

The Correct Option is B

The correct option is (B): \(-\displaystyle\int\limits^{\frac{1}{2}}_{0}\frac{In(1-t)}{t}dt.\)