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Chapter 9: Problem 22

Use the Root Test to determine whether the following series converge. $$\sum_{k=1}^{\infty}\left(1+\frac{3}{k}\right)^{k^{2}}$$

Short Answer

Expert verified
Answer: The Root Test is inconclusive for this series when applied, as the limit equals 1. Therefore, we cannot determine whether the series converges or diverges using the Root Test alone.

Step by step solution

01

Identify the given series

The given series is: $$\sum_{k=1}^{\infty}\left(1+\frac{3}{k}\right)^{k^{2}}$$ Where $$a_{k} = \left(1+\frac{3}{k}\right)^{k^{2}}$$.
02

Apply the Root Test

According to the Root Test, we need to find the limit $$\lim_{k\to\infty}\sqrt[k]{|a_{k}|}$$. In this case, we have: $$\lim_{k\to\infty}\sqrt[k]{\left|\left(1+\frac{3}{k}\right)^{k^{2}}\right|}$$
03

Simplify the expression

Notice that the expression under the absolute value sign is already positive, so the absolute value sign can be removed. Then, use the property of exponents inside the limit: $$\lim_{k\to\infty} \left(1+\frac{3}{k}\right)^{\frac{k^{2}}{k}}$$ The power simplifies to k: $$\lim_{k\to\infty} \left(1+\frac{3}{k}\right)^{k}$$
04

Evaluate the limit

As $$k\to\infty$$, the fraction $$\frac{3}{k}\to0$$. Hence, the limit becomes: $$\lim_{k\to\infty} \left(1+0\right)^{k} = \lim_{k\to\infty} 1^{k} = 1$$
05

Determine the convergence of the series

Since the limit from the Root Test is equal to 1, the test is inconclusive, meaning that it does not provide enough information about convergence or divergence. Therefore, we cannot determine whether the series converges or diverges using the Root Test only.

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Most popular questions from this chapter

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k} k}{2 k+1}$$

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Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty} \cos (\pi k)$$

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