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Chapter 10: Problem 14

Determine if the alternating series converges or diverges. Some of the series do not satisfy the conditions of the Alternating Series Test. $$\sum_{n=1}^{\infty}(-1)^{n+1} \frac{3 \sqrt{n+1}}{\sqrt{n}+1}$$

Short Answer

Expert verified
The series diverges because \(a_n\) does not converge to zero.

Step by step solution

01

Check Alternating Series Form

The given series is \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{3\sqrt{n+1}}{\sqrt{n}+1} \). This has an alternating sign component \((-1)^{n+1}\), which fits the form for an alternating series.
02

Define the Sequence Terms

Define \(a_n = \frac{3\sqrt{n+1}}{\sqrt{n}+1}\). We need to check if \(a_n\) satisfies the conditions of the Alternating Series Test. These conditions include that \(a_n\) should be positive, decreasing, and tends towards zero as \(n\) tends to infinity.
03

Check for Positivity

The sequence \(a_n = \frac{3\sqrt{n+1}}{\sqrt{n}+1}\) is positive for all \(n\geq1\) since the numerator and denominator are both positive. Thus, \(a_n > 0\) for all \(n\).
04

Check for Decreasing Nature

To determine if \(a_n\) is decreasing, we need to check if \(a_{n+1} < a_n\). We compute \(a_{n+1} = \frac{3\sqrt{n+2}}{\sqrt{n+1}+1}\) and compare it with \(a_n\). Algebraically simplifying or examining \(a_{n+1} - a_n\) would be necessary to verify this inequality.
05

Verify Limiting Behavior

Find \( \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{3\sqrt{n+1}}{\sqrt{n}+1} \). For large \(n\), this expression behaves similarly to \(\lim_{n \to \infty} \frac{3\sqrt{n}}{\sqrt{n}} = 3\). Since the limit of \(a_n\) is not zero, one of the conditions for convergence of the alternating series test is not met.

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Key Concepts

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

Series Convergence
Series convergence is all about determining whether the sum of an infinite series adds up to a finite number. In the context of alternating series, which are series with terms that switch signs between positive and negative, convergence can be tested using the Alternating Series Test. For an alternating series to converge, the sequence of absolute values of its terms must decrease monotonically to zero. If these conditions are met, the series will converge to a specific value even as the number of terms goes to infinity. If not, the series diverges, meaning its sum doesn't settle at any particular value.
Sequence Limits
Understanding sequence limits is crucial when discussing series convergence, especially in alternating series tests. A sequence limit is the value a sequence approaches as the number of terms goes to infinity. For example, if the sequence of terms in a series approaches zero as we consider more terms, it could imply the series converges. However, if the sequence doesn't tend towards zero, it indicates that the series may not converge. For the alternating series to converge, the limit of the sequence of terms must be zero. In our exercise, since the limit is not zero, it suggests divergence of the series.
Decreasing Sequence
A sequence is considered decreasing if each term is less than or equal to the preceding one. In mathematical analysis, verifying if a sequence of terms decreases is essential, especially for testing alternating series convergence. To apply this test correctly, it's necessary to determine that for each pair of consecutive terms, the later term is not greater than the previous one. This is done by checking conditions such as \( a_{n+1} < a_n \). However, if verifying algebraically becomes cumbersome, alternative approaches such as graphical or numerical analysis may need to be used to ensure or counter the decreasing condition.
Mathematical Analysis
Mathematical analysis provides the tools and methodologies needed to rigorously study concepts like series convergence and sequence behaviors. It involves examining limits, continuity, and differentiability, among other properties. When analyzing series, leveraging tools of mathematical analysis enables a more robust understanding of how series behave and whether they meet certain criteria for convergence. Analyzing the behavior of the sequence within the series, for example, allows estimation of limits and identifying trends like decreasing or increasing behaviors, facilitating a deeper understanding of series characteristics such as convergence or divergence.

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Most popular questions from this chapter

Use a CAS to perform the following steps for the sequences. a. Calculate and then plot the first 25 terms of the sequence. Does the sequence appear to be bounded from above or below? Does it appear to converge or diverge? If it does converge, what is the limit \(L\) ? b. If the sequence converges, find an integer \(N\) such that \(\left|a_{n}-L\right| \leq 0.01\) for \(n \geq N .\) How far in the sequence do you have to get for the terms to lie within 0.0001 of \(L ?\) \(a_{n}=\frac{8^{n}}{n !}\)

Assume that each sequence converges and find its limit. \(a_{1}=2, \quad a_{n+1}=\frac{72}{1+a_{n}}\)

Prove the "zipper theorem" for sequences: If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) both converge to \(L,\) then the sequence $$a_{1}, b_{1}, a_{2}, b_{2}, \dots, a_{n}, b_{n}, \dots$$ converges to \(L\).

Use Taylor's formula with \(n=2\) to find the quadratic approximation of \(f(x)=(1+x)^{k}\) at \(x=0\) ( \(k\) a constant). b. If \(k=3,\) for approximately what values of \(x\) in the interval [0,1] will the error in the quadratic approximation be less

The series $$\sin x=x-\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}-\frac{x^{7}}{7 !}+\frac{x^{9}}{9 !}-\frac{x^{11}}{11 !}+\cdots$$ converges to \(\sin x\) for all \(x\) a. Find the first six terms of a series for \(\cos x\). For what values of \(x\) should the series converge? b. By replacing \(x\) by \(2 x\) in the series for \(\sin x,\) find a series that converges to \(\sin 2 x\) for all \(x\) c. Using the result in part (a) and series multiplication, calculate the first six terms of a series for \(2 \sin x \cos x .\) Compare your answer with the answer in part (b).
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