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Q. 15

Expert-verifiedFound in: Page 692

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.

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

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.

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