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Chapter 8: Problem 48

Determine whether the series converges absolutely, converges conditionally, or diverges. The tests that have been developed in Section 5 are not the most appropriate for some of these series. You may use any test that has been discussed in this chapter. \(\sum_{n=1}^{\infty}(-3 / 4)^{n} n\)

Short Answer

Expert verified
The series converges absolutely.

Step by step solution

01

Understanding the series

We need to analyze the series \( \sum_{n=1}^{\infty}(-3 / 4)^{n} n \). This is a series where each term \( a_n = (-3/4)^n n \) is the product of an exponential and a linear factor.
02

Check if the series is alternating

Identify if the series is alternating by checking the presence of \((-1)^n\) or similar. The series \( (-3/4)^n \) includes the negative base, making this appear to be an alternating series. However, this series is not strictly alternating because \((-3/4)^n\) is not just \((-1)^n\) times a positive part, both the sign changes and amplitude decrease.
03

Apply the Absolute Convergence Test

To determine absolute convergence, examine \( \sum_{n=1}^{\infty} |(-3/4)^n n| \), simplifying to \( \sum_{n=1}^{\infty} (3/4)^n n \). We are checking if \( \sum_{n=1}^{\infty} (3/4)^n n \) converges.
04

Use the Ratio Test for Convergence

For the series \( \sum_{n=1}^{\infty} (3/4)^n n \), use the Ratio Test. Let \( a_n = (3/4)^n n \). Then, \( \frac{a_{n+1}}{a_n} = \frac{(3/4)^{n+1} (n+1)}{(3/4)^n n} = \frac{3}{4} \cdot \frac{n+1}{n} = \frac{3}{4} \cdot \left(1 + \frac{1}{n}\right) \).
05

Evaluate the Limit from the Ratio Test

Calculate \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \frac{3}{4} \cdot \lim_{n \to \infty}\left(1 + \frac{1}{n}\right) = \frac{3}{4} \). Since the limit is less than 1, the original series \( \sum_{n=1}^{\infty} (3/4)^n n \) converges.
06

Conclude the Type of Convergence

Since \( \sum_{n=1}^{\infty} |(-3/4)^n n| = \sum_{n=1}^{\infty} (3/4)^n n \) converges, the original series \( \sum_{n=1}^{\infty} (-3/4)^n n \) converges absolutely.

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Key Concepts

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

Absolute Convergence
Absolute convergence is an important concept when studying series. A series is said to converge absolutely if the series of the absolute values of its terms converges. Consider the series
  • If we have \(\sum_{n=1}^{\infty} a_n\), then the series converges absolutely if \(\sum_{n=1}^{\infty} |a_n|\) is convergent.
  • In our specific problem, we checked for absolute convergence by considering \(\sum_{n=1}^{\infty} |(-3/4)^n n|\), which simplifies to \(\sum_{n=1}^{\infty} (3/4)^n n\).
  • Since this new series was found to converge, it follows that the original series also converges absolutely.
Understanding whether a series converges absolutely is helpful as it guarantees the series is convergent regardless of how the terms behave regarding sign.
Ratio Test
The Ratio Test is a powerful tool for determining the convergence of a series. It is particularly useful for series with terms involving powers or factorials.
Given a series \(\sum_{n=1}^{\infty} a_n\), where \(a_n\) are non-zero, apply the ratio test as follows:
  • Calculate \(L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\).
  • If \(L < 1\), the series converges absolutely.
  • If \(L > 1\) or \(L = \infty\), the series diverges.
  • If \(L = 1\), the ratio test is inconclusive.
In our example, \(a_n = (3/4)^n n\), and the ratio test resulted in a limit \(L = 3/4 < 1\), indicating absolute convergence of the series.
Alternating Series
An alternating series is a series where the terms alternately take on positive and negative signs. The classic form of an alternating series \(\sum_{n=1}^{\infty} (-1)^n b_n\), where \(b_n > 0\).
  • The Alternating Series Test tells us an alternating series \(\sum (-1)^n b_n\) converges if \(b_n\) is decreasing and \(\lim_{n \to \infty} b_n = 0\).
  • While our problem had an element of sign change due to \(( -3/4 )^n\), it does not strictly fit the criteria for an alternating series \((e.g., lacks a clear \((-1)^n\) term).\).
Therefore, strictly applying alternating series tests wasn't directly suitable. Instead, other tests like absolute convergence can be considered.
Convergence Tests
Convergence tests give us tools to understand whether a series converges or not. This is crucial for determining the behavior and properties of series. Here's a snapshot of some common tests:
  • **Direct Comparison Test**: Compare your series with another series that is known to converge or diverge.
  • **Limit Comparison Test**: Involves taking the limit of the ratio between the terms of two series, providing a clearer picture of convergence in comparison to a known series.
  • **Integral Test**: Relates the series to a suitable integral to determine convergence.
  • **Ratio Test**: Used when terms of the series are defined with factorials or exponential functions.
  • **Root Test**: Another test that can be useful for series involving radicals.
All these tests help to understand whether a given series converges or diverges. Knowing how to apply these tests can optimize the approach to verify the convergence of unknown series.

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Most popular questions from this chapter

Consider the initial value problem $$ \frac{d y}{d x}=2-x-y, \quad y(0)=1 $$ \(\begin{array}{llll}\text { a. Calculate the power series } & \text { expansion }\end{array}\) \(y(x)=\sum_{n=0}^{\infty} a_{n} x^{n}\) of the solution up to the \(x^{7}\) term. b. Using the coefficients you have calculated, plot \(S_{3}(x)=\sum_{n=0}^{3} a_{n} x^{n}\) in the viewing rectangle \([-2,2] \times\) [-10,1.7] c. The exact solution to the initial value problem is \(y(x)=3-x-2 e^{-x},\) as can be determined using the methods of Section 7.7 (in Chapter 7 ). Add the plot of the exact solution to the viewing window. From the two plots, we see that the approximation is fairly accurate for \(-1 \leq x \leq 1\), but the accuracy decreases outside this subinterval. d. To see the improvement in accuracy that results from using more terms in a partial sum, replace the graph of \(S_{3}(x)\) with that of \(S_{7}(x)\)

Suppose that \(f^{\prime \prime}\) is continuous on an open interval centered at \(c .\) Use Taylor's Theorem to show that $$ \lim _{h \rightarrow 0} \frac{f(c+h)-2 f(c)+f(c-h)}{h^{2}}=f^{\prime \prime}(c) $$

In each of Exercises 49-54, use Taylor series to calculate the given limit. $$ \lim _{x \rightarrow 0} \frac{1-e^{2 x}}{x} $$

In each of Exercises 82-85, use an alternating Maclaurin series to approximate the given expression to four decimal places. $$ \exp (-0.2) $$

In each of Exercises 49-54, use Taylor series to calculate the given limit. $$ \lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}}{x} $$
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