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Chapter 9: Problem 56

Find the limit of the following sequences or state that they diverge. $$\left\\{\frac{\cos (n \pi / 2)}{\sqrt{n}}\right\\}$$

Short Answer

Expert verified
$$\lim_{n\to\infty} \frac{\cos (n \pi / 2)}{\sqrt{n}} = 0$$

Step by step solution

01

Observe the term \(\cos(n\pi/2)\)

Let's analyze the behavior of the cosine term first. Observe the values of \(\cos(n\pi/2)\) for different natural numbers n: - when n=1, \(\cos(1\pi/2) = \cos(\pi/2) = 0\) - when n=2, \(\cos(2\pi/2) = \cos(\pi) = -1\) - when n=3, \(\cos(3\pi/2) = \cos(3\pi/2) = 0\) - when n=4, \(\cos(4\pi/2) = \cos(2\pi) = 1\) - ... The values oscillate between 0, -1, 0, 1, and so on because the cosine function has a periodic behavior.
02

Observe the denominator term \(\sqrt{n}\)

Now, let's observe the denominator term \(\sqrt{n}\). As n goes to infinity, the value of \(\sqrt{n}\) tends to infinity as well.
03

Analyze the sequence as a whole

With the behavior of both terms in mind, we can now analyze the sequence as a whole: $$\frac{\cos (n \pi / 2)}{\sqrt{n}}$$ As n approaches infinity, the numerator oscillates between -1, 0, and 1. However, the denominator keeps growing to infinity.
04

Determine the limit of the sequence

We can now conclude that the sequence converges to 0 as n approaches infinity. This is because although the cosine term oscillates, the denominator grows larger, so for large values of n, the entire expression has a magnitude that gets closer and closer to 0. Therefore, the limit of the sequence is: <|ipynb_marker|> Code import sympy from sympy import cos, sqrt, pi, oo, simplify n = sympy.Symbol('n') sequence = cos(n*pi/2)/sqrt(n) sequence_limit = sympy.limit(sequence, n, oo) sequence_limit

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Most popular questions from this chapter

Consider the following sequences defined by a recurrence relation. Use a calculator, analytical methods, and/or graphing to make a conjecture about the value of the limit or determine that the limit does not exist. $$a_{n+1}=\frac{1}{2} a_{n}+2 ; a_{0}=5, n=0,1,2, \dots$$

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

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 \(10 ?\)

Given that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6},\) show that \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{2}}=\frac{\pi^{2}}{12}.\) (Assume the result of Exercise 63.)

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.
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