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Chapter 9: Problem 1

Explain why the sequence of partial sums for an alternating series is not an increasing sequence.

Short Answer

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5. This pattern continues, with odd-indexed partial sums being greater than their immediate predecessor (even-indexed) and even-indexed partial sums being less than their immediate predecessor (odd-indexed). From this observation, we can conclude that the sequence of partial sums for an alternating series is not increasing because the value of the partial sums alternates between increasing and decreasing as the terms are added or subtracted following the pattern. This behavior of alternating series makes the partial sums not strictly increasing or decreasing.

Step by step solution

01

Write down the general form of an alternating series

An alternating series can be represented by the general form: S = `鈭�((-1)^n)a_n` The series alternates between positive and negative terms, where `a_n` is a sequence of positive terms.
02

Observe the sequence of partial sums

To analyze the behavior of the alternating series, let's observe the sequence of its first few partial sums. The partial sums are given by: S_1 = `(-1)^0*a_0 = a_0` S_2 = `a_0 - a_1` S_3 = `a_0 - a_1 + a_2` S_4 = `a_0 - a_1 + a_2 - a_3` S_5 = `a_0 - a_1 + a_2 - a_3 + a_4` ... S_n = `鈭�((-1)^n)*a_n` for n = 1, 2, 3, ...
03

Understand the behavior of the partial sums

Let's analyze the behavior of the partial sums. By examining the partial sums, we can see that they follow a pattern: 1. The first partial sum S_1 is positive, as it equals to a_0. 2. The second partial sum S_2 has a value less than S_1 because a_1 is subtracted from a_0. 3. The third partial sum S_3 has a value greater than S_2 because a_2 is added to S_2. 4. The fourth partial sum S_4 has a value less than S_3 because a_3 is subtracted from S_3.

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Most popular questions from this chapter

Convergence parameter Find the values of the parameter \(p>0\) for which the following series converge. $$\sum_{k=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 k-1)}{p^{k} k !}$$

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3 / 2}}$$

Determine whether the following statements are true and give an explanation or counterexample. a. A series that converges must converge absolutely. b. A series that converges absolutely must converge. c. A series that converges conditionally must converge. d. If \(\sum a_{k}\) diverges, then \(\Sigma\left|a_{k}\right|\) diverges. e. If \(\sum a_{k}^{2}\) converges, then \(\sum a_{k}\) converges. f. If \(a_{k}>0\) and \(\sum a_{k}\) converges, then \(\Sigma a_{k}^{2}\) converges. g. If \(\Sigma a_{k}\) converges conditionally, then \(\Sigma\left|a_{k}\right|\) diverges.

The CORDIC (COordinate Rotation DIgital Calculation) algorithm is used by most calculators to evaluate trigonometric and logarithmic functions. An important number in the CORDIC algorithm, called the aggregate constant, is \(\prod_{n=0}^{\infty} \frac{2^{n}}{\sqrt{1+2^{2 n}}},\) where \(\prod_{n=0}^{N} a_{n}\) represents the product \(a_{0} \cdot a_{1} \cdots a_{N}\). This infinite product is the limit of the sequence $$\left\\{\prod_{n=0}^{0} \frac{2^{n}}{\sqrt{1+2^{2 n}}} \cdot \prod_{n=0}^{1} \frac{2^{n}}{\sqrt{1+2^{2 n}}}, \prod_{n=0}^{2} \frac{2^{n}}{\sqrt{1+2^{2 n}}} \ldots .\right\\}.$$ Estimate the value of the aggregate constant.

Given any infinite series \(\Sigma a_{k},\) let \(N(r)\) be the number of terms of the series that must be summed to guarantee that the remainder is less than \(10^{-r}\), where \(r\) is a positive integer. a. Graph the function \(N(r)\) for the three alternating \(p\) -series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{p}},\) for \(p=1,2,\) and \(3 .\) Compare the three graphs and discuss what they mean about the rates of convergence of the three series. b. Carry out the procedure of part (a) for the series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !}\) and compare the rates of convergence of all four series.
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