Question

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

Short answer

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

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 \(|x|<b\).

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: $$ f^{\prime}(x) = \frac{d}{dx}\left(\sum_{k=0}^{\infty} c_{k}x^{k}\right)= \sum_{k=0}^{\infty} \frac{d}{dx} \left(c_{k}x^{k}\right) $$

Calculate the term-by-term derivatives

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

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, $$ f^{\prime}(x)=\sum_{k=1}^{\infty} (c_{k} kx^{k-1}) $$

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.