Q. 10 - Calculus

Question

What is the relationship between a Maclaurin series and a power series in x?

Step-by-Step Solution

Answer

$$\begin{gathered} \mathrm{The} \mathrm{power} \mathrm{series} \mathrm{in} \mathrm{x} \mathrm{is} \mathrm{same} \mathrm{as} \mathrm{the} \mathrm{Maclaurin} \mathrm{series} \mathrm{in} \mathrm{x} \mathrm{if} {\mathrm{x}}_0=0 \\ \mathrm{is} \mathrm{considered}, \mathrm{that} \mathrm{is}, {\mathrm{a}}_{\mathrm{k}}=\frac{{\mathrm{f}}^{\mathrm{k}}\left(0\right)}{\mathrm{k}!}. \mathrm{The} \mathrm{Maclaurin} \mathrm{series} \mathrm{in} \mathrm{x} \mathrm{is} \mathrm{nothing} \mathrm{but} \mathrm{a} \\ \mathrm{special} \mathrm{case} \mathrm{of} \mathrm{power} \mathrm{series}, \mathrm{that} \mathrm{is}, \mathrm{a} \mathrm{power} \mathrm{series} \mathrm{calculated} \mathrm{about} \mathrm{origin} \mathrm{is} \\ \mathrm{the} \mathrm{Maclaurin} \mathrm{series} \mathrm{of} \mathrm{the} \mathrm{function}. \end{gathered}$$

Step 1. Given information is:

A Maclaurin series and a power series in x 

Step 2. Maclaurin series and Power series

$$\begin{gathered} \mathrm{The} \mathrm{power} \mathrm{series} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{f} \mathrm{around} \mathrm{the} \mathrm{point} \mathrm{x}={\mathrm{x}}_0 \mathrm{is}: \\ \mathrm{f}\left(\mathrm{x}\right) = \sum _{\mathrm{k}=0}^{\infty }\frac{{\mathrm{f}}^{\mathrm{k}}\left({\mathrm{x}}_0\right)}{\mathrm{k}!}{(\mathrm{x}-{\mathrm{x}}_0)}^{\mathrm{k}} \\ \mathrm{The} \mathrm{Maclaurin} \mathrm{series} \mathrm{of} \mathrm{the} \mathrm{function} \mathrm{f} \mathrm{around} \mathrm{the} \mathrm{point} \mathrm{x}=0 \mathrm{is}: \\ \mathrm{P}\left(\mathrm{x}\right)=\sum _{\mathrm{k}=0}^{\infty }\frac{{\mathrm{f}}^{\mathrm{k}}\left(0\right)}{\mathrm{k}!}{\left(\mathrm{x}\right)}^{\mathrm{k}} \end{gathered}$$

Step 3. Result

$$\begin{gathered} \mathrm{Therefore}, \mathrm{the} \mathrm{power} \mathrm{series} \mathrm{in} \mathrm{x} \mathrm{is} \mathrm{same} \mathrm{as} \mathrm{the} \mathrm{Maclaurin} \mathrm{series} \mathrm{in} \mathrm{x} \mathrm{if} {\mathrm{x}}_0=0 \\ \mathrm{is} \mathrm{considered}, \mathrm{that} \mathrm{is}, {\mathrm{a}}_{\mathrm{k}}=\frac{{\mathrm{f}}^{\mathrm{k}}\left(0\right)}{\mathrm{k}!}. \mathrm{The} \mathrm{Maclaurin} \mathrm{series} \mathrm{in} \mathrm{x} \mathrm{is} \mathrm{nothing} \mathrm{but} \mathrm{a} \\ \mathrm{special} \mathrm{case} \mathrm{of} \mathrm{power} \mathrm{series}, \mathrm{that} \mathrm{is}, \mathrm{a} \mathrm{power} \mathrm{series} \mathrm{calculated} \mathrm{about} \mathrm{origin} \mathrm{is} \\ \mathrm{the} \mathrm{Maclaurin} \mathrm{series} \mathrm{of} \mathrm{the} \mathrm{function}. \end{gathered}$$