Problem 5
Question
If \(f(x)=\sum_{k=0}^{\infty} c_{k} x^{k}\) and the series converges for
\(|x|
Step-by-Step Solution
Verified Answer
Answer: The power series representation of the derivative \(f^{\prime}(x)\) is given by \(\sum_{k=1}^{\infty} (c_{k} kx^{k-1})\).
1Step 1: 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|
2Step 2: 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:
$$
f^{\prime}(x) = \frac{d}{dx}\left(\sum_{k=0}^{\infty} c_{k}x^{k}\right)= \sum_{k=0}^{\infty} \frac{d}{dx} \left(c_{k}x^{k}\right)
$$
3Step 3: Calculate the term-by-term derivatives
Next, we will differentiate each term \(c_{k}x^{k}\):
$$
\frac{d}{dx} \left(c_{k}x^{k}\right) = c_{k} \frac{d}{dx} \left(x^{k}\right) = c_{k}(kx^{k-1})
$$
4Step 4: 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,
$$
f^{\prime}(x)=\sum_{k=1}^{\infty} (c_{k} kx^{k-1})
$$
5Step 5: 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|
Other exercises in this chapter
Problem 4
Explain why a power series is tested for absolute convergence.
View solution Problem 4
In general, how many terms do the Taylor polynomials \(p_{2}\) and \(p_{3}\) have in common?
View solution Problem 5
Suppose you know the Maclaurin series for \(f\) and it converges for \(|x|
View solution Problem 5
Do the interval and radius of convergence of a power series change when the series is differentiated or integrated? Explain.
View solution