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Chapter 2: Problem 132

Write each sum in sigma notation. \(1-a+a^{2}-a^{3}+a^{4}-a^{5}+\cdots+(-1)^{n} a^{n}\)

Short Answer

Expert verified
The sum is written as \(\sum_{k=0}^{n} (-1)^k a^k\).

Step by step solution

01

Identify the pattern

First, observe the terms in the given sum: \(1, -a, a^2, -a^3, a^4, -a^5, \ldots, (-1)^n a^n\). Notice that each term involves a power of \(a\) with alternating signs.
02

Determine the general term

The general term follows the form \((-1)^k a^k\). Here, \((-1)^k\) ensures the alternating sign, and \(a^k\) represents the power of \(a\) for each term. Each term of the sequence corresponds to each value of \(k\) from 0 to \(n\).
03

Write in sigma notation

Now that we understand the pattern, we can express the series in sigma notation as follows: \[\sum_{k=0}^{n} (-1)^k a^k\] This expression represents the sum of all terms from \(k = 0\) to \(k = n\) using the general term derived earlier.

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Key Concepts

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

Alternating Series
An alternating series is a sequence of numbers or terms where their signs switch between positive and negative. This sign change is crucial for understanding many mathematical concepts, especially in convergence of series. In the exercise, the sequence alternates signs with terms like 1, -\(a\), \(a^2\), and -\(a^3\).
  • The presence of \((-1)^k\) in a general term is typical for an alternating series. It switches the sign from positive to negative back again, as \((-1)^k\) becomes -1 or +1 depending on whether \(k\) is odd or even.
  • This alternating nature can have significant effects on the sum of the series, particularly in terms of convergence, which is a property describing whether a series approaches a finite limit as more terms are added.
Understanding alternating series is essential when analyzing more complex mathematical scenarios, as they often appear in calculus and analysis when dealing with series convergence.
Power Series
A power series is an infinite series that includes terms involving powers of a variable and is expressed in a specific form. It鈥檚 usually written as \(a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots\), where each term includes a coefficient and a variable raised to some power.
  • Each term increases the degree of the power by one compared to the previous term, a common characteristic seen in the given exercise with the powers of \(a\).
  • Power series have a radius of convergence, meaning that they converge within a certain interval around the center of the series. This is important to consider while using power series in practical applications such as function approximation and solution of differential equations.
  • In the example provided, though the series ends at \((-1)^n a^n\), it can be thought of as a finite power series, which contrasts to infinite power series commonly studied in calculus.
The concept of power series is fundamental in mathematical analysis, as it helps approximate functions, solve equations, and model natural phenomena.
General Term in a Sequence
The general term in a sequence is a formula that describes any given term in the series by specifying its position. It provides a systematic way to identify each element of the series.
  • In this exercise, the general term is noted as \((-1)^k a^k\). The \(k\) here indicates the position of the term within the sequence and can be substituted to derive each specific term.
  • Understanding the general term helps in writing and interpreting sequences in sigma notation, often simplifying complex problems by summarizing the entire series in a concise form.
  • This formula also aids in determining various properties of the series, like convergence and the behavior of series as the number of terms grows large.
Learning how to extract and understand the general term of a sequence is crucial when working with summations and series, as it simplifies the analysis and calculation processes.

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Most popular questions from this chapter

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.8, x_{0}=0.9\)

Write each sum in sigma notation. \(\frac{3}{5}+\frac{4}{6}+\frac{5}{7}+\frac{6}{8}+\frac{7}{9}\)

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Mountain Gorilla Conservation You are trying to build a mathematical model for the size of the population of mountain gorillas in a national park in Uganda. The data in this question are taken from Robbins et al. (2009). (a) You start by writing a word equation relating the population of gorillas \(t\) years after the study begins, \(N_{t}\), to the population \(N_{t+1}\) in the next year: \(N_{t+1}=N_{t}+\begin{array}{l}\text { number of gorillas } \\ \text { born in one year }\end{array}-\begin{array}{l}\text { number of gorillas } \\\ \text { that die in one yeai }\end{array}\) We will derive together formulas for the number of births and the number of deaths. (i) Around half of gorillas are female, \(75 \%\) of females are of reproductive age, and in a given year \(22 \%\) of the females of reproductive age will give birth. Explain why the number of births is equal to: \(\quad 0.5 \cdot 0.75 \cdot 0.22 \cdot N_{t}=0.0825 \cdot N_{t}\) (ii) In a given year \(4.5 \%\) of gorillas will die. Write down a formula for the number of deaths. (iii) Write down a recurrence equation for the number of gorillas in the national park. Assuming that there are 300 gorillas initially (that is \(N_{0}=300\) ), derive an explicit formula for the number of gorillas after \(t\) years. (iv) Calculate the population size after \(1,2,5\), and 10 years. (v) According to the model, how long will it take for population size to double to 600 gorillas? (b) In reality the population size is almost totally stagnant (i.e., \(N_{t}\) changes very little from year to year). Robbins et al. (2009) consider three different explanations for this effect: (i) Increased mortality: Gorillas are dying sooner than was thought. What percentage of gorillas would have to die each year for the population size to not change from year to year? (ii) Decreased female fecundity: Gorillas are having fewer offspring than was thought. Calculate the female birth rate (percentage of reproductive age females that give birth) that would lead to the population size not changing from year to year. Assume that all other values used in part (a) are correct. (iii) Emigration: Gorillas are leaving the national park. What number of gorillas would have to leave the national park each year for the population to not change from year to year?

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