Powers of x multiplied by a power series Prove that if

Chapter 8, Problem 63AE

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QUESTION:

Powers of x multiplied by a power series Prove that if \(f(x)=\sum_{k=0}^{\infty} c_{k} x^{k}\) converges on the interval I, then the power series for \(x^{m} f(x)\) also converges on I for positive integers m.

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QUESTION:

Powers of x multiplied by a power series Prove that if \(f(x)=\sum_{k=0}^{\infty} c_{k} x^{k}\) converges on the interval I, then the power series for \(x^{m} f(x)\) also converges on I for positive integers m.

ANSWER:

Solution 63AE

Step 1:

Given that

converges on the interval I

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