A function $f$ is defined by $$f(x) = \frac{1}{3} + \frac{2}{3^2}x + \frac{3}{3^3}x^2 + \cdots + \frac{n+1}{3^{n+1}}x^n + \cdots$$ for all $x$ in the interval of convergence of the given power series. (a) Find the interval of convergence for this power series. Show the work that leads to your answer. (b) Find $\displaystyle\lim_{x \rightarrow 0} \frac{f(x) - \frac{1}{3}}{x}$. (c) Write the first three nonzero terms and the general term for an infinite series that represents $\displaystyle\int_0^1 f(x)\, dx$. (d) Find the sum of the series determined in part (c).
A function $f$ is defined by
$$f(x) = \frac{1}{3} + \frac{2}{3^2}x + \frac{3}{3^3}x^2 + \cdots + \frac{n+1}{3^{n+1}}x^n + \cdots$$
for all $x$ in the interval of convergence of the given power series.
(a) Find the interval of convergence for this power series. Show the work that leads to your answer.
(b) Find $\displaystyle\lim_{x \rightarrow 0} \frac{f(x) - \frac{1}{3}}{x}$.
(c) Write the first three nonzero terms and the general term for an infinite series that represents $\displaystyle\int_0^1 f(x)\, dx$.
(d) Find the sum of the series determined in part (c).