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Chapter 9: Problem 24

Use the Root Test to determine whether the following series converge. $$\sum_{k=1}^{\infty}\left(\frac{1}{\ln (k+1)}\right)^{k}$$

Short Answer

Expert verified
Answer: Yes, the series converges.

Step by step solution

01

Define the sequence

First, let's define the sequence of the terms of the series as: $$ a_k = \left(\frac{1}{\ln (k+1)}\right)^{k}$$
02

Apply the Root Test

Now, we use the Root Test, which means we need to find the limit of the nth root of the absolute value of the terms of the series as n goes to infinity: $$\lim_{k \to \infty} \sqrt[k]{|a_k|}$$ In our case, the terms of the series are always positive, so we don't need to worry about the absolute value: $$\lim_{k \to \infty} \sqrt[k]{a_k} = \lim_{k \to \infty} \sqrt[k]{\left(\frac{1}{\ln (k+1)}\right)^{k}}$$
03

Simplify the limit

Now, we simplify the expression: $$\lim_{k \to \infty} \sqrt[k]{\left(\frac{1}{\ln (k+1)}\right)^{k}} = \lim_{k \to \infty} \frac{1}{\ln (k+1)}$$
04

Evaluate the limit

Evaluating the limit as k goes to infinity: $$\lim_{k \to \infty} \frac{1}{\ln (k+1)} = 0$$ Since the limit is less than 1, the series converges by the Root Test. Therefore, the series $$\sum_{k=1}^{\infty}\left(\frac{1}{\ln (k+1)}\right)^{k}$$ converges.

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