Problem 11 - Calculus Early Transcendentals

Question

Determine the radius of convergence of the following power series. Then test the endpoints to determine the interval of convergence. $$\sum \frac{(x-1)^{k}}{k}$$

Step-by-Step Solution

Answer

Answer: The radius of convergence is 1, and the interval of convergence is \(0 \le x < 2\).

Step 1: Apply the Ratio Test

To find the radius of convergence, let's apply the Ratio Test to the given power series. The Ratio Test formula is: $$\lim_{k \to \infty} \frac{\lvert a_{k+1} \rvert}{\lvert a_{k} \rvert} = L$$ The power series is given as: $$\sum \frac{(x-1)^{k}}{k}$$ To apply the Ratio Test, let's calculate the limit: $$L = \lim_{k \to \infty} \frac{\lvert \frac{(x-1)^{k+1}}{k+1} \rvert}{\lvert \frac{(x-1)^{k}}{k} \rvert}$$

Step 2: Simplify the limit expression

Now, let's simplify the limit expression, by canceling out terms: $$L = \lim_{k \to \infty} \frac{(k)(x-1)(x-1)^{k}}{(k+1)}$$ Now bring out the \(|(x-1)|\) term: $$L = |x-1| \lim_{k \to \infty} \frac{k}{k+1}$$

Step 3: Calculate the limit

To find the convergence, we'll calculate the limit: $$L = |x-1| \lim_{k \to \infty} \frac{1}{1+\frac{1}{k}}$$ The fraction inside the limit approaches 1 as \(k\) goes to infinity, so we have: $$L = |x-1|$$ For convergence, \(L < 1\), which is the condition for the ratio test. Thus, $$|x-1| < 1$$

Step 4: Find the radius of convergence

To find the radius of convergence \(R\), we'll solve the inequality obtained in step 3: $$-1 < x-1 < 1$$ Adding 1 to all parts of the inequality, we get: $$0 < x < 2$$ The radius of convergence, \(R\), is equal to half the length of the interval: $$R = \frac{2}{2} = 1$$ Thus, the radius of convergence is 1.

Step 5: Test the endpoints

Now we need to test the endpoints of the interval to determine the interval of convergence. For \(x = 0\), the series is \(\sum \frac{(-1)^k}{k}\), which is a convergent alternating series. For \(x = 2\), the series is \(\sum \frac{1^k}{k}\), which is a harmonic series that is known to be divergent.

Step 6: Write the interval of convergence

Based on the previous step, we can now write down the interval of convergence for the given power series: The interval of convergence is \(0 \le x < 2\).