Question

The power series expansion of a function is given as \[ \frac{1}{x}\ln(1 + x) = 1 + bx + cx^2 + ..... \] for $0<x \leq 1$. The values of constants $b$ and $c$, respectively, are

• A. $-\dfrac{1}{2}$ and $\dfrac{1}{3}$
• B. $\dfrac{1}{2}$ and $-\dfrac{1}{3}$
• C. $-1$ and $\dfrac{1}{2}$
• D. $1$ and $-\dfrac{1}{2}$

Hint:

The Taylor expansion of $\ln(1+x)$ is useful for finding the coefficients in power series expansions. Be sure to divide by $x$ when necessary.

Answer 1

The Correct Option is B

The given power series is the expansion of the function $\frac{1}{x} \ln(1 + x)$, which can be expanded as: \[ \ln(1 + x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - ..... \] Dividing this by $x$ gives the series: \[ \frac{1}{x} \ln(1 + x) = 1 + \dfrac{x}{2} - \dfrac{x^2}{3} + ..... \] Thus, the value of $b = \dfrac{1}{2}$ and $c = -\dfrac{1}{3}$.