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Chapter 9: Problem 2

Describe how to apply the Alternating Series Test.

Short Answer

Expert verified
Question: Determine if the following alternating series converges or diverges using the Alternating Series Test: \[ \sum_{n=1}^\infty (-1)^n \frac{n}{2n+1} \]

Step by step solution

01

Identify an alternating series

An alternating series is a series of the form: \[ \sum_{n=1}^\infty (-1)^{n-1}b_n \text{ or } \sum_{n=1}^\infty (-1)^n b_n, \] where \(b_n\) are non-negative terms. If the given series matches either of these forms, then it's an alternating series, and we can proceed with the Alternating Series Test.
02

Check the first condition - Decreasing terms

For the first condition, we need to check if the series has decreasing terms, i.e., \(b_{n+1} \leq b_n\) for all \(n\). In essence, the non-negative parts of the series should be decreasing or non-increasing.
03

Check the second condition - Limit of terms

For the second condition, we need to check if the limit of the terms, as \(n\) approaches infinity, is zero. That is, \[ \lim_{n\to\infty} b_n = 0. \] If this condition is met, then the terms are getting smaller and smaller as the series progresses.
04

Determine convergence or divergence

Once both conditions have been checked, if they both hold true, then the alternating series converges. If one or both of the conditions don't hold true, the Alternating Series Test is inconclusive, and another test may be needed to determine the convergence or divergence of the series. Note that the Alternating Series Test can only prove convergence, not divergence. If the test fails, it may still be possible to prove the divergence of the series using another test, such as the Comparison Test or the Ratio Test.

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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?

a. Consider the number 0.555555...., which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k} .\) Evaluate the geometric series to obtain a rational value of \(0.555555 \ldots\) b. Consider the number \(0.54545454 \ldots,\) which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k} .\) Evaluate the geometric series to obtain a rational value of the number. c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length \(p,\) say, \(n_{1}, n_{2} \ldots \ldots, n_{p},\) where \(n_{1}, \ldots, n_{p}\) are integers between 0 and \(9 .\) Explain how to use geometric series to obtain a rational form of the number. d. Try the method of part (c) on the number \(0.123456789123456789 \ldots\) e. Prove that \(0 . \overline{9}=1\)

Convergence parameter Find the values of the parameter \(p>0\) for which the following series converge. $$\sum_{k=2}^{\infty} \frac{1}{(\ln k)^{p}}$$

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{k}}{k^{2}}$$

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