Problem 5 - Calculus Early Transcendentals

Question

Do the interval and radius of convergence of a power series change when the series is differentiated or integrated? Explain.

Step-by-Step Solution

Answer

In summary, the radius of convergence remains the same when a power series is differentiated or integrated. However, the interval of convergence may change at the endpoints, as differentiating or integrating can affect the convergence of the series at those points.

Step 1: Write the general power series

Consider a general power series in x: $$ \sum_{n=0}^{\infty} a_n (x - c)^n $$ where \(a_n\) are constant coefficients, and \(c\) is the center of the series.

Step 2: Find the radius of convergence using the ratio test

We use the ratio test for convergence to determine the radius of convergence \(R\): $$ R = \lim_{n\to\infty} \left|\frac{a_n}{a_{n+1}}\right| $$

Step 3: Differentiate the power series term-by-term

Now we differentiate the power series term-by-term: $$ \frac{d}{dx}\left( \sum_{n=0}^{\infty} a_n (x - c)^n \right) = \sum_{n=1}^{\infty} n a_n (x - c)^{n-1} $$

Step 4: Find the radius of convergence of the differentiated series using the ratio test

Now let's apply the ratio test to the differentiated series: $$ R' = \lim_{n\to\infty} \left|\frac{n a_n}{(n+1) a_{n+1}}\right| = \lim_{n\to\infty} \left|\frac{a_n}{a_{n+1}}\right| $$ As we can see, the radius of convergence remains the same after differentiation, as \(R' = R\).

Step 5: Integrate the power series term-by-term

Now we integrate the power series term-by-term: $$ \int \left(\sum_{n=0}^{\infty} a_n (x - c)^n \right) dx = \sum_{n=0}^{\infty} \frac{a_n}{n+1} (x - c)^{n+1} + C $$

Step 6: Find the radius of convergence of the integrated series using the ratio test

Now let's apply the ratio test to the integrated series: $$ R'' = \lim_{n\to\infty} \left|\frac{(n+1) a_n}{(n+2) a_{n+1}}\right| = \lim_{n\to\infty} \left|\frac{a_n}{a_{n+1}}\right| $$ As we can see, the radius of convergence remains the same after integration, as \(R'' = R\).

Step 7: Answer and Explanation

The radius of convergence does not change when a power series is differentiated or integrated. The interval of convergence may change, however, at the endpoints of the interval, since differentiating or integrating can affect the convergence of the series at those endpoints.