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Chapter 11: Problem 7

\(2-28\) Determine whether the series is absolutely convergent, conditionally convergent, or divergent. $$\sum_{k=1}^{\infty} k\left(\frac{2}{3}\right)^{k}$$

Short Answer

Expert verified
The series is absolutely convergent.

Step by step solution

01

Check for Absolute Convergence

To determine if the series is absolutely convergent, we first examine the absolute value of the terms:\[\sum_{k=1}^{\infty} \left|k \left(\frac{2}{3}\right)^{k} \right| = \sum_{k=1}^{\infty} k \left(\frac{2}{3}\right)^{k}\]Since this series is the same as the original series, we will assess the convergence of this series to determine absolute convergence.
02

Apply the Ratio Test

The ratio test can be used to determine convergence. Consider the ratio of consecutive terms:\[\lim_{{k \to \infty}} \left| \frac{(k+1)\left(\frac{2}{3}\right)^{k+1}}{k\left(\frac{2}{3}\right)^{k}} \right|\]Simplifying gives:\[\lim_{{k \to \infty}} \left( k+1 \right) \frac{2}{3} \cdot \frac{1}{k}\]\[\lim_{{k \to \infty}} \frac{k+1}{k} \cdot \frac{2}{3} = \frac{2}{3}\]The limiting value is \(\frac{2}{3} < 1\), so the series converges absolutely by the ratio test.

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Key Concepts

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

Ratio Test
The Ratio Test is a powerful tool to determine the convergence or divergence of an infinite series. It works by examining the ratio of consecutive terms in the series. Here's how you can apply the test:
  • Take your series with the terms represented by \(a_k\).
  • Form the ratio of the \((k+1)\)th term and the \(k\)th term, \(\frac{a_{k+1}}{a_k}\).
  • Compute the limit: \(\lim_{{k \to \infty}} \left|\frac{a_{k+1}}{a_k}\right|\).
If this limit is less than 1, the series converges absolutely. If it's greater than 1, the series is divergent. And if it equals 1, the test is inconclusive. In the given exercise, using the ratio test helped determine absolute convergence by showing that the limit is \(\frac{2}{3}\), which is less than 1. This helps confirm the series converges absolutely.
Series Convergence
Convergence of a series involves determining whether the sum of terms in the series approaches a finite number as more terms are added. When we say a series is convergent, it implies that as you add more terms, the total approaches a specific value.There are different types of convergence:
  • Absolute Convergence: This occurs when the series of absolute values, \(\sum |a_k|\), converges. It usually implies strong convergence properties.
  • Conditional Convergence: A series converges conditionally if it converges but does not do so absolutely. This means that the series \(\sum a_k\) converges while \(\sum |a_k|\) does not.
In our exercise, the term \(k\left(\frac{2}{3}\right)^k\) was analyzed for absolute convergence using the ratio test, affirming that the series converges absolutely.
Infinite Series
An infinite series is essentially the sum of an infinite list of numbers, written in the form \(\sum_{k=1}^{\infty} a_k\). It is a fundamental concept in mathematical analysis and calculus. Infinite series can exhibit various convergence behaviors, and understanding where and how they converge is essential.Types of infinite series include:
  • Geometric Series: Has a constant ratio between successive terms. The series \(a + ar + ar^2 + \ldots\) converges if the absolute value of the ratio \(r\) is less than 1.
  • Arithmetic Series: Each term is a constant difference from its predecessor. These do not typically converge as they progress indefinitely.
  • P-Series: Takes the form \(\sum \frac{1}{n^p}\). Converges when \(p > 1\).
In the exercise, our focus was on a modified geometric-type series involving a variable multiplier, \(k\). The infinite nature of the series is critical in applying tests like the ratio test for convergence, ultimately helping determine the behavior of the series across its infinite span.

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Most popular questions from this chapter

Find the sum of the series. \(1-\ln 2+\frac{(\ln 2)^{2}}{2 !}-\frac{(\ln 2)^{3}}{3 !}+\cdots\)

Let \(a\) and \(b\) be positive numbers with \(a > b .\) Let \(a_{1}\) be their arithmetic mean and \(b_{1}\) their geometric mean: $$ a_{1}=\frac{a+b}{2} \quad b_{1}=\sqrt{a b} $$ Repeat this process so that, in general, $$ a_{n+1}=\frac{a_{n}+b_{n}}{2} \quad b_{n+1}=\sqrt{a_{n} b_{n}} $$ (a) Use mathematical induction to show that $$ a_{n}>a_{n+1}>b_{n+1}>b_{n} $$ (b) Deduce that both \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are convergent. (c) Show that \(\lim _{n \rightarrow \infty} a_{n}=\lim _{n \rightarrow \infty} b_{n} .\) Gauss called the common value of these limits the arithmetic-geometric mean of the numbers a and \(b\) .

Let $$f(x)=\sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}}$$ Find the intervals of convergence for \(f, f^{\prime},\) and \(f^{\prime \prime}\)

The resistivity \(\rho\) of a conducting wire is the reciprocal of the conductivity and is measured in units of ohm-meters \((\Omega-m) .\) The resistivity of a given metal depends on the temperature according to the equation $$\rho(t)=\rho_{20} e^{\alpha(t-20)}$$ where \(t\) is the temperature in \(^{\circ} \mathrm{C}\) . where \(t\) is the temperature in \(^{\circ} \mathrm{C} .\) There are tables that list the values of \(\alpha\) (called the temperature coefficient) and \(\rho_{20}\) (the resistivity at \(20^{\circ} \mathrm{C} )\) for various metals. Except at very low temperatures, the resistivity varies almost linearly with temperature and so it is common to approximate the expression for \(\rho(t)\) by its first- or second-degree Taylor polynomial at \(t=20\) . (a) Find expressions for these linear and quadratic approximations. (b) For copper, the tables give \(\alpha=0.0039 /^{\circ} \mathrm{C}\) and \(\rho_{20}=1.7 \times 10^{-8} \Omega-\mathrm{m} .\) Graph the resistivity of copper and the linear and quadratic approximations for \(-250^{\circ} \mathrm{C} \leqslant t \leqslant 1000^{\circ} \mathrm{C}\) (c) For what values of \(t\) does the linear approximation agree with the exponential expression to within one percent?

Test the series for convergence or divergence. $$\sum_{n=1}^{\infty} \frac{n^{2}+1}{5^{n}}$$
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