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Chapter 9: Problem 63

Define an alternating series and state the Alternating Series Test.

Short Answer

Expert verified
An alternating series is a series whose terms alternate in sign. The Alternating Series Test states that if the terms of an alternating series are non-increasing and the limit as \( n \) approaches infinity is zero, then the series converges.

Step by step solution

01

Define Alternating Series

An alternating series is a series whose terms alternate in sign. More specifically, it is a series of the form \( a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + ... \) or \( -a_1 + a_2 - a_3 + a_4 - a_5 + a_6 - ...\), where, for all \( n \), \( a_n \) is a positive real number.
02

State the Alternating Series Test

The Alternating Series Test is criterion used to check whether an alternating series converges or diverges. It states that if an alternating series fulfills two conditions - 1) The terms \( a_n \) are non-increasing, that is \( a_{n+1} \leq a_n \) for every \( n \), and 2) The limit of the terms as \( n \) approaches infinity is zero, that is \( lim_{n \to \infty} a_n = 0 \), then the series converges.

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Key Concepts

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

Alternating Series
Imagine a sequence of numbers where each term switches from positive to negative, much like a zigzag. This is the essence of an alternating series. In mathematics, an alternating series formally refers to a sum where the terms consistently switch signs. An example of such a series is \( a_1 - a_2 + a_3 - a_4 + a_5 - a_6 + ... \), where each \( a_n \) stands for a positive term.

Why do we care? In the realm of infinite series, determining whether a sum will eventually reach a stable value鈥攄espite having infinitely many terms鈥攊s critical. Alternating series can converge, which means they settle on a specific number, or diverge, meaning they fail to settle at all. The behavior of these intriguing series is crucial for understanding phenomena in physics, engineering, and beyond!
Series Convergence
Series convergence is a fundamental concept in calculus, describing the behavior of infinite series. The big question we ask is, 'Does this series add up to a finite number, despite having an infinite number of terms?' If an infinite series approaches a specific value as more and more terms are added, we say the series converges.

To determine convergence, various tests can be applied, depending on the series type. These tests are essential tools for mathematicians and scientists, enabling them to draw conclusions about series that often represent complex real-world situations.
Non-Increasing Sequence
In the study of sequences and series, the nature of sequence terms' progression is vital. A non-increasing sequence is one in which each term is less than or equal to the previous one. In other words, as we move from left to right in the sequence, the terms do not go up in value. This is expressed mathematically as \( a_{n+1} \leq a_n \) for all \( n \).

Understanding whether a sequence is non-increasing can be a deciding factor in determining a series' behavior鈥攕uch as convergence or divergence. This property helps impose a kind of order on what might otherwise be a chaotic and unruly infinite set of terms!
Limit of a Sequence
The concept of the limit of a sequence is at the heart of mathematical analysis. It answers the question: 'As we travel along the sequence towards infinity, does it approach a specific value?' The limit of a sequence is denoted as \( \lim_{{n \to \infty}} a_n \), which translates to the value that the terms \( a_n \) get closer and closer to, as \( n \) becomes very large.

For an alternating series to converge, one requirement is that the limit of its terms must be zero as \( n \) tends to infinity. This condition ensures that the 'tail end' of the series鈥攖he terms that come later and later鈥攂ecome insignificant and does not affect the overall sum dramatically, allowing convergence to a finite value.

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Most popular questions from this chapter

Use the formula for the \(n\) th partial sum of a geometric series \(\sum_{i=0}^{n-1} a r^{i}=\frac{a\left(1-r^{n}\right)}{1-r}\) Annuities When an employee receives a paycheck at the end of each month, \(P\) dollars is invested in a retirement account. These deposits are made each month for \(t\) years and the account earns interest at the annual percentage rate \(r\). If the interest is compounded monthly, the amount \(A\) in the account at the end of \(t\) years is $$ \begin{aligned} A &=P+P\left(1+\frac{r}{12}\right)+\cdots+P\left(1+\frac{r}{12}\right)^{12 t-1} \\ &=P\left(\frac{12}{r}\right)\left[\left(1+\frac{r}{12}\right)^{12 t}-1\right] \end{aligned} $$ If the interest is compounded continuously, the amount \(A\) in the account after \(t\) years is $$ \begin{aligned} A &=P+P e^{r / 12}+P e^{2 r / 12}+P e^{(12 t-1) r / 12} \\ &=\frac{P\left(e^{n}-1\right)}{e^{r / 12}-1} \end{aligned} $$ Verify the formulas for the sums given above.

The Fibonacci sequence is defined recursively by \(a_{n+2}=a_{n}+a_{n+1}\), where \(a_{1}=1\) and \(a_{2}=1\). (a) Show that \(\frac{1}{a_{n+1} a_{n+3}}=\frac{1}{a_{n+1} a_{n+2}}-\frac{1}{a_{n+2} a_{n+3}}\). (b) Show that \(\sum_{n=0}^{\infty} \frac{1}{a_{n+1} a_{n+3}}=1\).

Marketing An electronic games manufacturer producing a new product estimates the annual sales to be 8000 units. Each year \(10 \%\) of the units that have been sold will become inoperative. So, 8000 units will be in use after 1 year, \([8000+0.9(8000)]\) units will be in use after 2 years, and so on. How many units will be in use after \(n\) years?

The series represents a well-known function. Use a computer algebra system to graph the partial sum \(S_{10}\) and identify the function from the graph. $$ f(x)=\sum_{n=0}^{\infty}(-1)^{n} x^{n}, \quad-1

Evaluate the binomial coefficient using the formula \(\left(\begin{array}{l}k \\ n\end{array}\right)=\frac{k(k-1)(k-2)(k-3) \cdot \cdots(k-n+1)}{n !}\) where \(k\) is a real number, \(n\) is a positive integer, and \(\left(\begin{array}{l}k \\ 0\end{array}\right)=1\) \(\left(\begin{array}{c}0.5 \\ 4\end{array}\right)\)
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