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Chapter 9: Problem 15

Use the Ratio Test to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{2^{k}}{k^{99}}$$

Short Answer

Expert verified
Question: Determine if the series $$\sum_{k=1}^{\infty} \frac{2^{k}}{k^{99}}$$ converges using the Ratio Test. Answer: The series $$\sum_{k=1}^{\infty} \frac{2^{k}}{k^{99}}$$ diverges.

Step by step solution

01

Identify the terms of the series

The terms of the series can be represented by the general term $$a_k = \frac{2^k}{k^{99}}$$
02

Apply the Ratio Test by finding the limit

Using the Ratio Test, we need to find the limit as k approaches infinity of the absolute ratio of consecutive terms, which is given by: $$\lim_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right| = \lim_{k\to\infty} \frac{\frac{2^{k+1}}{(k+1)^{99}}}{\frac{2^k}{k^{99}}}$$
03

Simplify the expression

Simplify the expression by eliminating common factors: $$\lim_{k\to\infty} \frac{2^{k+1}}{(k+1)^{99}}\cdot\frac{k^{99}}{2^k} = \lim_{k\to\infty} \frac{2}{\left(\frac{k+1}{k}\right)^{99}}$$
04

Use limit properties to evaluate the expression

Using the limit properties, we get $$\lim_{k\to\infty} \frac{2}{\left(\frac{k+1}{k}\right)^{99}} = \frac{2}{\left(1+\frac{1}{k}\right)^{99}}$$ Since the denominator goes to \(1^{\infty}\) as k approaches infinity, the limit of the absolute ratio of consecutive terms is: $$\lim_{k\to\infty} \frac{2}{\left(1+\frac{1}{k}\right)^{99}}=2$$
05

Compare the limit with 1

Since the calculated limit is greater than 1, $$\lim_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right|=2 > 1$$
06

Conclude about the convergence

With the Ratio Test, when the limit is greater than 1, we can conclude that the series diverges. Therefore, the series $$\sum_{k=1}^{\infty} \frac{2^{k}}{k^{99}}$$ diverges.

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Most popular questions from this chapter

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. The Consumer Price Index (the CPI is a measure of the U.S. cost of living) is given a base value of 100 in the year \(1984 .\) Assume the CPI has increased by an average of \(3 \%\) per year since \(1984 .\) Let \(c_{n}\) be the CPI \(n\) years after \(1984,\) where \(c_{0}=100.\)

Determine whether the following statements are true and give an explanation or counterexample. a. A series that converges must converge absolutely. b. A series that converges absolutely must converge. c. A series that converges conditionally must converge. d. If \(\sum a_{k}\) diverges, then \(\Sigma\left|a_{k}\right|\) diverges. e. If \(\sum a_{k}^{2}\) converges, then \(\sum a_{k}\) converges. f. If \(a_{k}>0\) and \(\sum a_{k}\) converges, then \(\Sigma a_{k}^{2}\) converges. g. If \(\Sigma a_{k}\) converges conditionally, then \(\Sigma\left|a_{k}\right|\) diverges.

Use Exercise 89 to determine how many terms of each series are needed so that the partial sum is within \(10^{-6}\) of the value of the series (that is, to ensure \(R_{n}<10^{-6}\) ). a. \(\sum_{k=0}^{\infty} 0.72^{k}\) b. \(\sum_{k=0}^{\infty}(-0.25)^{k}\)

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