Question

If m is a positive integer, how can we find the Maclaurin series for the function $c{x}^{m}f\left(x\right)$if we already know the Maclaurin series for the function f(x)? How do you find the interval of convergence for the new series?

Short answer

The required value is$c{x}^{m}f\left(x\right)=\sum _{k=0}^{\infty }c{a}_k{x}^{k+m}$and interval of convergence will be same.

Step-by-step solution

Step 1. Given Information 

The given function is$c{x}^{m}f\left(x\right)$

Step 2. Explanation 

If the Maclaurin series for the function $f\left(x\right)=\sum _{k=0}^{\infty }{a}_k{x}^k$then the Maclaurin series for the function $c{x}^{m}f\left(x\right)$can be found by simply multiplying the maclaurin series of function by $c{x}^m$.

$$\begin{gathered} c{x}^{m}f\left(x\right)=c{x}^m\sum _{k=0}^{\infty }{a}_k{x}^k \\ =\underset{k=0}{\overset{\infty }{\sum c}}{a}_k{x}^{k+m} \end{gathered}$$

Also, the interval of convergence for two series would be same.