Q. 11 - Calculus

Question

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

Step-by-Step Solution

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 1. Given information is:

A function f with Maclaurin series

Step 2. Solving for interval

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