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Chapter 9: Problem 33

Use the Comparison Test or Limit Comparison Test to determine whether the following series converge. $$\sum_{k=1}^{\infty} \frac{\sin (1 / k)}{k^{2}}$$

Short Answer

Expert verified
Answer: Yes, the series converges.

Step by step solution

01

Identify the parent series

To use the Limit Comparison Test, we need to identify a parent series that we can compare with the given series. In this case, we can use the series \(\sum_{k=1}^{\infty} \frac{1}{k^2}\), which is a p-series with p=2. We know that a p-series converges if and only if p > 1, and in this case, p=2, so \(\sum_{k=1}^{\infty} \frac{1}{k^2}\) converges.
02

Compute the limit

Now we need to compute the limit of the ratio of the given series to the parent series. The limit we are interested in is: $$ \lim_{k \to \infty} \frac{\frac{\sin(1/k)}{k^2}}{\frac{1}{k^2}}. $$
03

Simplify the limit expression

Now, we can simplify the limit expression: $$ \lim_{k \to \infty} \frac{\frac{\sin(1/k)}{k^2}}{\frac{1}{k^2}} = \lim_{k \to \infty} \frac{\sin(1/k)}{1}. $$
04

Evaluate the limit

Evaluate the limit as k goes to infinity: $$ \lim_{k \to \infty} \frac{\sin(1/k)}{1} = \lim_{k \to \infty} \sin\left(\frac{1}{k}\right). $$ As k approaches infinity, the fraction \(\frac{1}{k}\) approaches 0. Hence, we can rewrite the limit as: $$ \lim_{k \to \infty} \sin\left(\frac{1}{k}\right) = \sin(0) = 0. $$
05

Apply the Limit Comparison Test

Since we found that the limit is 0 and since the parent series converges, we conclude that the given series also converges, according to the Limit Comparison Test. $$ \sum_{k=1}^{\infty} \frac{\sin (1 / k)}{k^{2}} $$ converges.

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Most popular questions from this chapter

A fishery manager knows that her fish population naturally increases at a rate of \(1.5 \%\) per month, while 80 fish are harvested each month. Let \(F_{n}\) be the fish population after the \(n\) th month, where \(F_{0}=4000\) fish. a. Write out the first five terms of the sequence \(\left\\{F_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{F_{n}\right\\}\). c. Does the fish population decrease or increase in the long run? d. Determine whether the fish population decreases or increases in the long run if the initial population is 5500 fish. e. Determine the initial fish population \(F_{0}\) below which the population decreases.

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k} k^{2}}{\sqrt{k^{6}+1}}$$

Consider the expression \(\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}},\) where the process continues indefinitely. a. Show that this expression can be built in steps using. the recurrence relation \(a_{0}=1, a_{n+1}=\sqrt{1+a_{n}},\) for \(n=0,1,2,3, \ldots .\) Explain why the value of the expression can be interpreted as \(\lim _{n \rightarrow \infty} a_{n}\). b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\). c. Estimate the limit of the sequence. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. e. Repeat the preceding analysis for the expression \(\sqrt{p+\sqrt{p+\sqrt{p+\sqrt{p+\cdots}}}}\) where \(p > 0 .\) Make a table showing the approximate value of this expression for various values of \(p .\) Does the expression seem to have a limit for all positive values of \(p ?\)

a. Sketch the function \(f(x)=1 / x\) on the interval \([1, n+1]\) where \(n\) is a positive integer. Use this graph to verify that $$ \ln (n+1)<1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}<1+\ln n $$ b. Let \(S_{n}\) be the sum of the first \(n\) terms of the harmonic series, so part (a) says \(\ln (n+1)0,\) for \(n=1,2,3, \ldots\) c. Using a figure similar to that used in part (a), show that $$ \frac{1}{n+1}>\ln (n+2)-\ln (n+1) $$ d. Use parts (a) and (c) to show that \(\left\\{E_{n}\right\\}\) is an increasing sequence \(\left(E_{n+1}>E_{n}\right)\) e. Use part (a) to show that \(\left\\{E_{n}\right\\}\) is bounded above by 1 f. Conclude from parts (d) and (e) that \(\left\\{E_{n}\right\\}\) has a limit less than or equal to \(1 .\) This limit is known as Euler's constant and is denoted \(\gamma\) (the Greek lowercase gamma). g. By computing terms of \(\left\\{E_{n}\right\\},\) estimate the value of \(\gamma\) and compare it to the value \(\gamma \approx 0.5772 .\) (It has been conjectured, but not proved, that \(\gamma\) is irrational.) h. The preceding arguments show that the sum of the first \(n\) terms of the harmonic series satisfy \(S_{n} \approx 0.5772+\ln (n+1)\) How many terms must be summed for the sum to exceed \(10 ?\)

Evaluate the limit of the following sequences. $$a_{n}=\frac{7^{n}}{n^{7} 5^{n}}$$
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