Short Answer

If m is a positive integer, how can we find the Maclaurin series for the function $c x^{m} f (x)$ 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 $c x^{m} f (x) = \sum_{k = 0}^{\infty} c a_{k} x^{k + m}$ and interval of convergence will be same.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

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

Step 2. Explanation

If the Maclaurin series for the function $f (x) = \sum_{k = 0}^{\infty} a_{k} x^{k}$ then the Maclaurin series for the function $c x^{m} f (x)$ can be found by simply multiplying the maclaurin series of function by $c x^{m}$ .

$c x^{m} f (x) = c x^{m} \sum_{k = 0}^{\infty} a_{k} x^{k} = {\sum c}_{k = 0}^{\infty} a_{k} x^{k + m}$

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

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Short Answer

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

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