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Chapter 9: Problem 13

Use the Root Test to determine whether each series converges absolutely or diverges. $$\sum_{n=1}^{\infty} \frac{-8}{(3+(1 / n))^{2 n}}$$

Short Answer

Expert verified
The series converges absolutely.

Step by step solution

01

Understand the Root Test

The Root Test states that for a series \( \sum a_n \), you consider \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). If this limit is less than 1, the series converges absolutely. If it is greater than 1, the series diverges. If it equals 1, the test is inconclusive.
02

Identify the Term \( a_n \)

For the given series \( \sum_{n=1}^{\infty} \frac{-8}{(3+(1 / n))^{2 n}} \), we can identify the term \( a_n = \frac{-8}{(3+(1 / n))^{2 n}} \). We will use its absolute value, which is \( |a_n| = \frac{8}{(3+(1 / n))^{2 n}} \).
03

Find \( \sqrt[n]{|a_n|} \)

Compute the n-th root of the absolute value of \( a_n \):\[ \sqrt[n]{|a_n|} = \sqrt[n]{\frac{8}{(3+(1 / n))^{2 n}}} = \frac{8^{1/n}}{3+(1/n)^2} \]
04

Calculate the Limit as \( n \to \infty \)

Evaluate the limit of \( \frac{8^{1/n}}{3+(1/n)^2} \) as \( n \to \infty \):- As \( n \to \infty \), \( 8^{1/n} \to 1 \) because any constant to the power of \( 1/n \) approaches 1.- \( 1/n \) approaches 0, so \( 3+(1/n)^2 \to 3 \).Thus, the limit is \( \lim_{n \to \infty} \frac{8^{1/n}}{3+(1/n)^2} = \frac{1}{3} \).
05

Examine the Limit in Terms of the Root Test

Since \( \frac{1}{3} < 1 \), the Root Test indicates that the series \( \sum_{n=1}^{\infty} \frac{-8}{(3+(1 / n))^{2 n}} \) converges absolutely.

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Key Concepts

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

Convergence
In the context of series, convergence is an essential concept. When we say a series converges, we mean that the sum of its terms approaches a specific value as we add more and more terms. Think of it like this: if you keep adding very small numbers indefinitely, does the total settle to a certain number? If yes, then the series converges.
  • A converging series implies stability - there鈥檚 a number it鈥檚 aiming for and reaching, no matter how far into the series you count.
  • The stability of a series is an indicator of its behavior - understanding whether a series converges or diverges tells us a lot about its characteristics.
Grasping the concept of convergence is key to deciphering many mathematical scenarios. For example, the series of all numbers appearing in a specific order might either settle into a regular pattern (convergence) or just keep growing indefinitely without settling (divergence).
Series
A series is the sum of the terms of a sequence. Imagine you have a list of numbers, and you're asked to add them all up; this process defines a series. But not all series add up to a meaningful value.
  • Some series accumulate to a finite number, indicating they are convergent and hold real-world applications like calculating compound interest or predicting weather patterns.
  • Other series diverge; they grow without bound or do not settle to a fixed number.
  • Understanding how to test series for convergence helps in various fields, from engineering to economics.
The Root Test you learned in the solution helps determine whether a series converges absolutely or not. Its application involves taking a closer look at the terms of the series, usually involving complex or unheard sequences which require careful analysis and powerful mathematical tools.
Absolute Convergence
Absolute convergence is a stronger form of convergence. If a series converges absolutely, then by considering the absolute values of its terms, the series remains convergent. This concept is more robust than standard convergence.
  • Absolute convergence assures that even if there's some alternating negativity and positivity in the series, stripping away the signs still leaves us with a convergent series.
  • In mathematics, a series that converges absolutely provides a form of certainty - it tells us that convergence is not dependent on how the signs of the terms affect the sum.
The Root Test, as applied in the exercise, confirmed the given series converges absolutely. Using absolute values gives mathematicians a clearer picture of a series' full behavior, showing that under no circumstances does it diverge.
Limit
The concept of a limit is foundational in calculus and analysis. It refers to the value that a function (or a sequence, or series) "approaches" as the input or index approaches some value. In the context of the Root Test, calculating the limit was crucial.
  • As described in the solution, finding the limit of a specific expression \( \sqrt[n]{|a_n|} \) was needed to apply the Root Test effectively.
  • The result of this limit was used to determine the absolute convergence of the series.
  • Limits help us tackle infinite processes with tangible methods, enabling predictions about the eventual behavior of sequences and series.
Understanding limits facilitate better comprehension of how elements behave when continuously approached. Whether it鈥檚 determining convergence, handling infinite sums, or delving into continuity, limits are central to understanding deeper mathematical phenomena.

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Most popular questions from this chapter

Assume that the series \(\sum a_{n} x^{n}\) converges for \(x=4\) and diverges for \(x=7 .\) Answer true (T), false (F), or not enough information given (N) for the following statements about the series. a. Converges absolutely for \(x=-4\) b. Diverges for \(x=5\) c. Converges absolutely for \(x=-8.5\) d. Converges for \(x=-2\) e. Diverges for \(x=8\) f. Diverges for \(x=-6\) g. Converges absolutely for \(x=0\) h. Converges absolutely for \(x=-7.1\)

The sum of the series \(\sum_{n=0}^{\infty}\left(n^{2} / 2^{n}\right)\) To find the sum of this series, express \(1 /(1-x)\) as a geometric series, differentiate both sides of the resulting equation with respect to \(x,\) multiply both sides of the result by \(x\), differentiate again, multiply by \(x\) again, and set \(x\) equal to \(1 / 2 .\) What do you get?

Use the Taylor series for \(1 /\left(1-x^{2}\right)\) to obtain a series for \(2 x /\left(1-x^{2}\right)^{2}\)

In the alternating harmonic series, suppose the goal is to arrange the terms to get a new series that converges to \(-1 / 2 .\) Start the new arrangement with the first negative term, which is \(-1 / 2 .\) Whenever you have a sum that is less than or equal to \(-1 / 2,\) start introducing positive terms, taken in order, until the new total is greater than \(-1 / 2 .\) Then add negative terms until the total is less than or equal to \(-1 / 2\) again. Continue this process until your partial sums have been above the target at least three times and finish at or below it. If \(s_{n}\) is the sum of the first \(n\) terms of your new series, plot the points \(\left(n, s_{n}\right)\) to illustrate how the sums are behaving.

Use any method to determine whether the series converges or diverges. Give reasons for your answer. $$\sum_{n=2}^{\infty} \frac{n}{(\ln n)^{(n / 2)}}$$
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