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Chapter 9: Problem 27

Determine whether the following series converge. $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{\sqrt{k^{2}+4}}$$

Short Answer

Expert verified
Based on the Alternating Series Test and the step-by-step solution provided, determine if the following series converges or diverges: $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{\sqrt{k^{2}+4}}$$ Answer: The series converges.

Step by step solution

01

Identify the alternating series

The given series can be written as: $$\sum_{k=0}^{\infty} (-1)^k a_k$$ where \(a_k = \frac{1}{\sqrt{k^2 + 4}}\) We can see that this is an alternating series because it has the factor \((-1)^k\).
02

Verify the first condition of the Alternating Series Test

The first condition of the Alternating Series Test is that the sequence \({a_k}\) must be non-increasing. We need to show that \(a_k \ge a_{k+1}\) for all \(k\). Since \(a_k = \frac{1}{\sqrt{k^2 + 4}}\), let's compare \(a_k\) and \(a_{k+1}\): $$ \frac{1}{\sqrt{k^2 + 4}} \ge \frac{1}{\sqrt{(k+1)^2 + 4}} $$ As \(k\) increases, the expression under the square root also increases, which makes the denominator larger and the whole fraction smaller. So, the condition is met; the sequence \({a_k}\) is non-increasing.
03

Verify the second condition of the Alternating Series Test

The second condition of the Alternating Series Test is that \(\lim_{k\to\infty} a_k = 0\). We need to show that this holds for our sequence. We have \(a_k = \frac{1}{\sqrt{k^2 + 4}}\). Let's find the limit as \(k\) goes to infinity: $$ \lim_{k\to\infty} \frac{1}{\sqrt{k^2 + 4}} $$ As \(k\) goes to infinity, the expression \(k^2 + 4\) also goes to infinity, making the denominator infinitely large and the whole fraction approaches zero: $$ \lim_{k\to\infty} \frac{1}{\sqrt{k^2 + 4}} = 0 $$ Thus, the second condition is met.
04

Conclude using the Alternating Series Test

Since both conditions of the Alternating Series Test have been met: 1. The sequence \({a_k}\) is non-increasing. 2. \(\lim_{k\to\infty} a_k = 0\). We can conclude that the given alternating series converges: $$\sum_{k=0}^{\infty} \frac{(-1)^{k}}{\sqrt{k^{2}+4}}$$ converges.

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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. $$a_{n+1}=2 a_{n}\left(1-a_{n}\right) ; a_{0}=0.3, n=0,1,2, \dots$$

Suppose a function \(f\) is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty} x^{2 k}\) a. Evaluate \(f(0), f(0.2), f(0.5), f(1),\) and \(f(1.5),\) if possible. b. What is the domain of \(f ?\)

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=0}^{\infty} x^{k}$$

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.)

The expression where the process continues indefinitely, is called a continued fraction. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=1+1 / a_{n},\) for \(n=0,1,2,3, \ldots . .\) Explain why the value of the expression can be interpreted as \(\lim a_{n}\). b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\). c. Using computation and/or graphing, estimate the limit of the sequence. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. e. Assuming the limit exists, use the same ideas to determine the value of where \(a\) and \(b\) are positive real numbers.
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