Q. 12 - Calculus

Question

If a function f has a Taylor series at $x_0$, 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 Taylor series at $x_0$

Step 2. Solving for interval

$$\begin{gathered} \mathrm{The} \mathrm{taylor} \mathrm{series} \mathrm{for} \mathrm{the} \mathrm{function} \mathrm{f} \mathrm{around} \mathrm{x}={\mathrm{x}}_0 \mathrm{is}: \\ \mathrm{P}\left({\mathrm{x}}_0\right)=\sum _{\mathrm{k}=0}^{\infty }\frac{{\mathrm{f}}^{\mathrm{k}}\left(0\right)}{\mathrm{k}!}{(\mathrm{x}-{\mathrm{x}}_0)}^{\mathrm{k}} \\ \mathrm{The} \mathrm{possible} \mathrm{value} \mathrm{for} \mathrm{the} \mathrm{interval} \mathrm{of} \mathrm{convergence} \mathrm{for} \mathrm{the} \mathrm{series} \mathrm{are} \\ (-\mathrm{\rho },\mathrm{\rho }),(-\mathrm{\rho },\mathrm{\rho }],[-\mathrm{\rho },\mathrm{\rho }) \mathrm{and} [-\mathrm{\rho },\mathrm{\rho }], \mathrm{here}, \mathrm{\rho }>0. \end{gathered}$$