MATH$152$ Lab $12$

Power Series Representations

This lab explores the process of finding a power series representation for a function and understanding what this representation means.

Problem $1$

Consider the function

$f(x)={\displaystyle \underset{n=1}{\overset{}{}}}\frac{x^n}{n^2}$

The radius of convergence for $f(x)$ is $1.$ A beautiful theorem guarantees that the radius of convergence for $f(x),f^{\text{'}}(x),$ and $f^{\text{'}\text{'}}(x)$ is the same, but the interval of convergence can change. Find the interval of convergence for $f(x),f^{\text{'}}(x),$ and $f^{\text{'}\text{'}}(x).$ Support your responses with the necessary analysis.

Interval of convergence for $f(x)$ :

Interval of convergence for $f^{\text{'}}(x)$ :

Interval of convergence for $f^{\text{'}\text{'}}(x)$ :