• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! Q. 15

Expert-verified Found in: Page 692 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # 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?

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.

See the 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 $c{x}^{m}$.

$c{x}^{m}f\left(x\right)=c{x}^{m}\sum _{k=0}^{\infty }{a}_{k}{x}^{k}\phantom{\rule{0ex}{0ex}}=\underset{k=0}{\overset{\infty }{\sum c}}{a}_{k}{x}^{k+m}$

Also, the interval of convergence for two series would be same. ### Want to see more solutions like these? 