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Chapter 10: Problem 11

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

Short Answer

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

Step by step solution

01

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: $$L = \lim_{k \to \infty} \frac{\lvert \frac{(x-1)^{k+1}}{k+1} \rvert}{\lvert \frac{(x-1)^{k}}{k} \rvert}$$
02

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: $$L = |x-1| \lim_{k \to \infty} \frac{k}{k+1}$$
03

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, $$|x-1| < 1$$
04

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.
05

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.
06

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 \(0 \le x < 2\).

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