Chapter 8: Q. 5 (page 692)

If the series $\sum_{k = 0}^{\infty} c_{k} {(x - x_{0})}^{k}$ converges to the function $h (x)$ for every real number, provide a formula for $c_{k}$ in terms of the function $h$ .

Short Answer

Expert verified

The formula for $c_{k}$ is $\frac{h^{(k)} (x_{0})}{k!}$ .

Step by step solution

Step 1. Given Information.

The series $\sum_{k = 0}^{\infty} c_{k} {(x - x_{0})}^{k}$ converges to $h (x)$ for every real number.

Step 2. Explanation.

The series is converges to the function so, Taylor polynomial for h(x) in the form

$\sum_{k = 0}^{\infty} \frac{h^{(k)} (x_{0})}{k!} {(x - x_{0})}^{k}$ .

For series $\sum_{k = 0}^{\infty} c_{k} {(x - x_{0})}^{k}$ ,role="math" localid="1649601140285" $c_{k} = \frac{h^{(k)} (x_{0})}{k!}$

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If the series $\sum_{k = 0}^{\infty} c_{k} {(x - x_{0})}^{k}$ converges to the function $h (x)$ for every real number, provide a formula for $c_{k}$ in terms of the function $h$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

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Step 1. Given Information.

Step 2. Explanation.

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If the series $\sum_{k = 0}^{\infty} c_{k} {(x - x_{0})}^{k}$ converges to the function $h (x)$ for every real number, provide a formula for $c_{k}$ in terms of the function $h$ .