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Ch. 10 - Infinite Sequences and Series
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 10 - Infinite Sequences and SeriesProblem 10.1.15
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

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1
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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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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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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Related Practice
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 = 1 to ∞) [ (ln n) xⁿ ]

Textbook Question

In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.

∑ (from n = 0 to ∞) [ (x² − 1) / 2 ]ⁿ

Textbook Question

Make up an infinite series of nonzero terms whose sum is

b. −3

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

1
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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