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Chapter 10: Problem 5

If \(f(x)=\sum_{k=0}^{\infty} c_{k} x^{k}\) and the series converges for \(|x|

Short Answer

Expert verified
Answer: The power series representation of the derivative \(f^{\prime}(x)\) is given by \(\sum_{k=1}^{\infty} (c_{k} kx^{k-1})\).

Step by step solution

01

Identify the power series representation of the function f(x)

In this exercise, we are given the power series representation of the function \(f(x)\) as: $$ f(x)=\sum_{k=0}^{\infty} c_{k}x^{k} $$ where \(c_k\) are the coefficients of the power series and \(x\) is a variable. This series converges for \(|x|<b\).
02

Differentiate the power series term-by-term

To find the power series representation of the derivative \(f^{\prime}(x)\), we differentiate the given power series term-by-term with respect to \(x\). Recall that the derivative of \(x^k\) with respect to \(x\) is \(kx^{k-1}\). Using this, we can differentiate each term of the power series: $$ f^{\prime}(x) = \frac{d}{dx}\left(\sum_{k=0}^{\infty} c_{k}x^{k}\right)= \sum_{k=0}^{\infty} \frac{d}{dx} \left(c_{k}x^{k}\right) $$
03

Calculate the term-by-term derivatives

Next, we will differentiate each term \(c_{k}x^{k}\): $$ \frac{d}{dx} \left(c_{k}x^{k}\right) = c_{k} \frac{d}{dx} \left(x^{k}\right) = c_{k}(kx^{k-1}) $$
04

Write the power series for the term-by-term derivatives

Now that we have found the derivative for each term in the power series, we can rewrite the series for the derivative \(f^{\prime}(x)\) as: $$ f^{\prime}(x)=\sum_{k=0}^{\infty} (c_{k} kx^{k-1}) $$ Notice that when k=0, the term will be zero (as the derivative of a constant term is zero). So we change the index of summation to start at k=1, $$ f^{\prime}(x)=\sum_{k=1}^{\infty} (c_{k} kx^{k-1}) $$
05

Write the final expression for the power series of the derivative

The power series representation of the derivative \(f^{\prime}(x)\) is: $$ f^{\prime}(x)=\sum_{k=1}^{\infty} (c_{k}kx^{k-1}) $$ This series will also converge for \(|x|<b\) since term-wise differentiation doesn't affect the interval of convergence.

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Most popular questions from this chapter

Find the function represented by the following series and find the interval of convergence of the series. $$\sum_{k=0}^{\infty}\left(x^{2}+1\right)^{2 k}$$

a. Find a power series for the solution of the following differential equations. b. Identify the function represented by the power series. $$y^{\prime}(t)+4 y(t)=8, y(0)=0$$

Use properties of power series, substitution, and factoring of constants to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. Use the Taylor series. $$(1+x)^{-2}=1-2 x+3 x^{2}-4 x^{3}+\cdots, \text { for }-1 < x < 1$$ $$(1+4 x)^{-2}$$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The Taylor polynomials for \(f(x)=e^{-2 x}\) centered at 0 consist of even powers only. b. For \(f(x)=x^{5}-1,\) the Taylor polynomial of order 10 centered at \(x=0\) is \(f\) itself. c. The \(n\) th-order Taylor polynomial for \(f(x)=\sqrt{1+x^{2}}\) centered at 0 consists of even powers of \(x\) only.

What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than \(10^{-3}\) ? (The answer depends on your choice of a center.) $$e^{-0.5}$$
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