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Chapter 8: Problem 9

Use the Divergence Test to determine whether the following series diverge or state that the test is inconclusive. $$\sum_{k=0}^{\infty} \frac{k}{2 k+1}$$

Short Answer

Expert verified
Answer: The series diverges.

Step by step solution

01

Identify the general term of the given series

The given series is \(\sum_{k=0}^{\infty} \frac{k}{2 k+1}\). Here, the general term (n-th term) of the series is given by \(a_k = \frac{k}{2 k+1}\).
02

Find the limit of the general term as it tends to infinity

We need to find the limit of the general term \(a_k\) as \(k\) tends to infinity. We can write it as: $$\lim_{k\to\infty} \frac{k}{2 k+1}$$
03

Simplify the expression for the limit

To find the limit of the given expression, we can divide both the numerator and the denominator by the highest power of \(k\) present in the expression. In this case, the highest power of \(k\) is 1. So, we divide both the numerator and the denominator by \(k\). Doing this, we get: $$\lim_{k\to\infty} \frac{\frac{k}{k}}{\frac{2 k}{k}+\frac{1}{k}}$$ Now, simplify the expression: $$\lim_{k\to\infty} \frac{1}{2+\frac{1}{k}}$$
04

Calculate the limit

Now, as \(k\) tends to infinity, the expression \(\frac{1}{k}\) tends to zero. So, the limit becomes: $$\lim_{k\to\infty} \frac{1}{2+0}$$ Therefore, the limit is: $$\lim_{k\to\infty} \frac{k}{2 k+1} = \frac{1}{2}$$
05

Apply the Divergence Test

The Divergence Test states that if the limit of the general term as it tends to infinity is not equal to zero, then the series diverges. In our case, $$\lim_{k\to\infty} \frac{k}{2 k+1} = \frac{1}{2} \neq 0$$ Since the limit is not equal to zero, we can conclude that the given series diverges.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Infinite Series
An infinite series is the sum of the terms in an infinite sequence. It extends indefinitely, adding up all the elements of the sequence one after another. Imagine you have an endless list of numbers that you want to add up. That's your infinite series! For example, if you have a series \( \sum_{k=0}^{\infty} a_k \), it means you are summing every term from \( a_0 \) to \( a_k \) as \( k \) approaches infinity.

Understanding infinite series is crucial in various fields of mathematics and science. They help in calculating complex phenomena, such as calculating the area under a curve, solving differential equations, and even in computer algorithms.
  • Elements of a Series: The series is built from its general or n-th term, \( a_n \), which is defined by a specific formula for each term.
  • Notation: The summation symbol \( \sum \) is used, which represents the sum of all terms from an initial index to an infinite point.
Grasping the basics of infinite series is essential for delving deeper into topics like convergence and divergence.
Convergence and Divergence
Whether an infinite series converges or diverges depends on the behavior of its terms as you progress along the series. Convergence occurs when the sum of a series approaches a fixed value, while divergence happens when the sum grows without bound or oscillates without settling on a value.

For convergence, it's essential that the terms are diminishing suitably fast as indicated by several tests like the Divergence Test, Ratio Test, and others. The Divergence Test, in particular, states that if the limit of the series' general term does not equal zero, the series diverges.
  • Convergent Series: In a convergent series, the total sum approaches a finite value as more terms are added.
  • Divergent Series: In a divergent series, the sum either increases without bounds or does not settle into a specific value.
  • Significance of Divergence Test: It is one of the simplest tools to determine divergence. If \( \lim_{n \to \infty} a_n eq 0 \), then the series is divergent.
Understanding these concepts helps in various analyses where predicting the behavior of an infinite sum is crucial.
Limit of a Sequence
The limit of a sequence is a core concept when dealing with infinite series. It refers to the value that the terms of a sequence approach as the index (usually represented by \( n \) or \( k \)) gets indefinitely large. Limits are fundamental in determining the behavior of sequences and infinite series. They allow us to decide whether a series converges or diverges.

To calculate the limit of a sequence, simplify the sequence's general term and observe what value it approaches as the index becomes very large. In the case of the series \( \sum_{k=0}^{\infty} \frac{k}{2k+1} \), the limit of the term is \( \frac{1}{2} \), which means that each term approaches this value as \( k \) goes to infinity.
  • Limit Expression: It is often denoted as \( \lim_{n \to \infty} a_n \), showing what value (if any) the terms are approaching.
  • Importance of Zero Limit: For a series to possibly converge, the limit of its terms must be zero. Otherwise, the series is immediately divergent without further tests.
Understanding limits is essential as they tell us whether our series might settle down to a specific sum or not.

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Most popular questions from this chapter

Here is a fascinating (unsolved) problem known as the hailstone problem (or the Ulam Conjecture or the Collatz Conjecture). It involves sequences in two different ways. First, choose a positive integer \(N\) and call it \(a_{0} .\) This is the seed of a sequence. The rest of the sequence is generated as follows: For \(n=0,1,2, \ldots\) $$a_{n+1}=\left\\{\begin{array}{ll} a_{n} / 2 & \text { if } a_{n} \text { is even } \\ 3 a_{n}+1 & \text { if } a_{n} \text { is odd .} \end{array}\right.$$ However, if \(a_{n}=1\) for any \(n,\) then the sequence terminates. a. Compute the sequence that results from the seeds \(N=2,3\), \(4, \ldots, 10 .\) You should verify that in all these cases, the sequence eventually terminates. The hailstone conjecture (still unproved) states that for all positive integers \(N\), the sequence terminates after a finite number of terms. b. Now define the hailstone sequence \(\left\\{H_{k}\right\\},\) which is the number of terms needed for the sequence \(\left\\{a_{n}\right\\}\) to terminate starting with a seed of \(k\). Verify that \(H_{2}=1, H_{3}=7\), and \(H_{4}=2\). c. Plot as many terms of the hailstone sequence as is feasible. How did the sequence get its name? Does the conjecture appear to be true?

The Greeks solved several calculus problems almost 2000 years before the discovery of calculus. One example is Archimedes' calculation of the area of the region \(R\) bounded by a segment of a parabola, which he did using the "method of exhaustion." As shown in the figure, the idea was to fill \(R\) with an infinite sequence of triangles. Archimedes began with an isosceles triangle inscribed in the parabola, with area \(A_{1}\), and proceeded in stages, with the number of new triangles doubling at each stage. He was able to show (the key to the solution) that at each stage, the area of a new triangle is \(\frac{1}{8}\) of the area of a triangle at the previous stage; for example, \(A_{2}=\frac{1}{8} A_{1},\) and so forth. Show, as Archimedes did, that the area of \(R\) is \(\frac{4}{3}\) times the area of \(A_{1}\).

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{4^{n}+5 n !}{n !+2^{n}}$$

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 \(c_{0}=100\)

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{75^{n-1}}{99^{n}}+\frac{5^{n} \sin n}{8^{n}}$$
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