Question

If m is a positive integer, how can we find the Maclaurin series for the function $f\left(c{x}^m\right)$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 $f\left(c{x}^m\right)=\sum _{k=0}^{\infty }{a}_k{c}^k{x}^{mk}$and the interval of convergence for the two series will be same.

Step-by-step solution

Step 1. Given Information 

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

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 $f\left(c{x}^m\right)$can be found by simply substituting $xbyc{x}^m$in the Maclaurin series of the function.

Thus,

$f\left(c{x}^m\right)=\sum _{k=0}^{\infty }{a}_k{\left(c{x}^m\right)}^{k}f\left(c{x}^m\right)=\sum _{k=0}^{\infty }{a}_k{c}^k{x}^{mk}$

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