Textbook Question
Make up an infinite series of nonzero terms whose sum is
b. −3
Ch. 10 - Infinite Sequences and Series
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 10, Problem 10.1.15
Finding a Sequence’s Formula
In Exercises 13–30, find a formula for the nth term of the sequence.
1, -4, 9, -16, 25, …Squares of the positive integers, with alternating signs
Verified step by step guidance1
Identify the pattern in the sequence: the terms are squares of positive integers with alternating signs. The sequence is 1, -4, 9, -16, 25, ... which corresponds to 1^2, -2^2, 3^2, -4^2, 5^2, ...
Express the nth term as the square of n, which is \(n^2\).
Incorporate the alternating sign. Since the signs alternate starting with positive for n=1, use \((-1)^{n+1}\) to represent the sign pattern: positive when n is odd, negative when n is even.
Combine the sign and the square to write the general formula for the nth term as \(a_n = (-1)^{n+1} \times n^2\).
Verify the formula by plugging in the first few values of n to ensure it matches the given sequence.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sequences and Terms
A sequence is an ordered list of numbers where each number is called a term. The nth term represents the value at position n in the sequence. Understanding how to express the nth term as a formula allows you to find any term without listing all previous terms.
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Guided course
8:22
Introduction to Sequences
Square Numbers
Square numbers are integers raised to the power of two, expressed as n². In this sequence, each term’s absolute value corresponds to the square of its position index, such as 1², 2², 3², and so on, which helps identify the pattern in the sequence.
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4:47The Number e
Alternating Signs
Alternating signs mean the terms switch between positive and negative values in a regular pattern. This can be represented mathematically using (-1)ⁿ or (-1)ⁿ⁺¹, which changes the sign of each term depending on whether n is even or odd.
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10:54Alternating Series Test
Related Practice
Textbook Question
Repeating Decimals
Express each of the numbers in Exercises 23–30 as the ratio of two integers.
3.1̅4̅2̅8̅5̅7 = 3.142857142857 ...
Textbook Question
In Exercises 37–42, find the series’ radius of convergence.
∑ (from n = 1 to ∞) [ ( n / (n + 1) )ⁿ^ ² ] xⁿ (Hint: Apply the Root Test.)
Textbook Question
Which series in Exercises 53–76 converge, and which diverge? Give reasons for your answers. If a series converges, find its sum.
∑ (from n = 1 to ∞) (1 − 1/n)ⁿ
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Textbook Question
Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?
∑ (from n = 0 to ∞) (x + 5)ⁿ
Textbook Question
Finding Taylor and Maclaurin Series
In Exercises 25–34, find the Taylor series generated by f at x = a.
f(x) = x³ − 2x + 4,a = 2
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