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Chapter 9: Problem 13

Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. $$\sum_{k=0}^{\infty} \frac{1}{1000+k}$$

Short Answer

Expert verified
Answer: The Divergence Test is inconclusive for the given series.

Step by step solution

01

Find the limit of the sequence

To find the limit of the sequence, we will take the limit as k approaches infinity: $$\lim_{k\to\infty} \frac{1}{1000+k}$$
02

Calculate the limit

As k approaches infinity, the denominator increases without bound. Therefore, the entire fraction approaches zero: $$\lim_{k\to\infty} \frac{1}{1000+k} = 0$$
03

Apply the Divergence Test

Since the limit of the sequence equals zero, the Divergence Test is inconclusive. This means we cannot determine if the series converges or diverges using this test alone. In summary, the Divergence Test is inconclusive for the given series: $$\sum_{k=0}^{\infty} \frac{1}{1000+k}$$

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Most popular questions from this chapter

Suppose a ball is thrown upward to a height of \(h_{0}\) meters. Each time the ball bounces, it rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nth bounce and let \(S_{n}\) be the total distance the ball has traveled at the moment of the nth bounce. a. Find the first four terms of the sequence \(\left\\{S_{n}\right\\}\) b. Make a table of 20 terms of the sequence \(\left\\{S_{n}\right\\}\) and determine a plausible value for the limit of \(\left\\{S_{n}\right\\}.\) $$h_{0}=20, r=0.5$$

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. $$\sum_{k=1}^{\infty} \cos (\pi k)$$

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. $$\sum_{k=1}^{\infty} 3^{-k}$$

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty}\left(-\frac{1}{3}\right)^{k}$$

Convergence parameter Find the values of the parameter \(p>0\) for which the following series converge. $$\sum_{k=2}^{\infty} \frac{\ln k}{k^{p}}$$
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