Question

If a function f has a Maclaurin series, what are the possibilities for the interval of convergence for that series?

Short answer

$$\begin{gathered} \mathrm{The}\mathrm{possible}\mathrm{value}\mathrm{for}\mathrm{the}\mathrm{interval}\mathrm{of}\mathrm{convergence}\mathrm{for}\mathrm{the}\mathrm{series}\mathrm{are} \\ (-\mathrm{\rho },\mathrm{\rho }),(-\mathrm{\rho },\mathrm{\rho }],[-\mathrm{\rho },\mathrm{\rho })\mathrm{and}[-\mathrm{\rho },\mathrm{\rho }],\mathrm{here},\mathrm{\rho }>0. \end{gathered}$$

Step-by-step solution

Step 1. Given information is:

A function f with Maclaurin series

Step 2. Solving for interval

$$\begin{gathered} \mathrm{If}\mathrm{f}\mathrm{has}\mathrm{a}\mathrm{Maclaurin}\mathrm{series},\mathrm{that}\mathrm{is}, \\ {\mathrm{P}}_{\mathrm{n}}\left(\mathrm{x}\right)=\sum _{\mathrm{k}=0}^{\infty }\frac{{\mathrm{f}}^{\mathrm{k}}\left(0\right)}{\mathrm{k}!}{\left(\mathrm{x}\right)}^{\mathrm{k}} \\ \mathrm{The}\mathrm{possible}\mathrm{value}\mathrm{for}\mathrm{the}\mathrm{interval}\mathrm{of}\mathrm{convergence}\mathrm{for}\mathrm{the}\mathrm{series}\mathrm{are} \\ (-\mathrm{\rho },\mathrm{\rho }),(-\mathrm{\rho },\mathrm{\rho }],[-\mathrm{\rho },\mathrm{\rho })\mathrm{and}[-\mathrm{\rho },\mathrm{\rho }],\mathrm{here},\mathrm{\rho }>0. \end{gathered}$$