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Chapter 10: Problem 74

Find the values of \(x\) for which the given geometric series converges. Also, find the sum of the series (as a function of \(x\) ) for those values of \(x\). $$\sum_{n=0}^{\infty}(-1)^{n} x^{-2 n}$$

Short Answer

Expert verified
The series converges for \(|x| > 1\); sum is \(\frac{x^2}{x^2 + 1}\).

Step by step solution

01

Identify the Series

The given series is \( \sum_{n=0}^{\infty}(-1)^{n} x^{-2n} \). It is a geometric series where each term has the form \( ar^n \). Here, the first term \( a = 1 \), and the common ratio \( r = (-1) x^{-2} \).
02

Conditions for Convergence

A geometric series \( \sum_{n=0}^{\infty} ar^n \) converges if \( |r| < 1 \). For our series, \( r = (-1) x^{-2} \), so we need \( |-1 \cdot x^{-2}| < 1 \). This simplifies to solving \( |x|^{-2} < 1 \), which further simplifies to \( |x| > 1 \).
03

Conclusion on Convergence

The series converges for values of \( x \) such that \( |x| > 1 \). This means \( x > 1 \) or \( x < -1 \).
04

Calculate Sum of the Series

When a geometric series converges, its sum is given by \( \frac{a}{1 - r} \). Using \( a = 1 \) and \( r = (-1) x^{-2} \), the sum is \( \frac{1}{1 - (-1) x^{-2}} \).
05

Simplify the Sum Expression

The expression for the sum becomes \( \frac{1}{1 + x^{-2}} \). This can be rewritten as \( \frac{x^2}{x^2 + 1} \), which is the sum of the series for \( |x| > 1 \).

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Key Concepts

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

Convergence
Convergence is when a series approaches a specific value, even as the number of terms increases infinitely. For geometric series, convergence depends on the common ratio, denoted as \( r \).
The series will only converge if the absolute value of \( r \) is less than 1. In this problem, the geometric series is \( \sum_{n=0}^{\infty}(-1)^{n} x^{-2n} \).
  • Here, the common ratio \( r = (-1) x^{-2} \).
  • To ensure convergence, we need \( |-1 \times x^{-2}| < 1 \).
  • This simplifies to \( |x|^{-2} < 1 \), further implying \( |x| > 1 \).
Thus, the series converges for \( x \) values such that \( |x| > 1 \), meaning \( x > 1 \) or \( x < -1 \).
Sum of Series
When a geometric series is convergent, it has a sum that can be calculated. The formula to find the sum \( S \) of a convergent series \( \sum_{n=0}^{\infty} ar^n \) is given by \( S = \frac{a}{1 - r} \).
In our series, the first term \( a = 1 \), and the common ratio \( r = (-1) x^{-2} \).
  • Start by inserting these values into the formula: \( S = \frac{1}{1 - (-1)x^{-2}} \).
  • This simplifies to \( \frac{1}{1 + x^{-2}} \).
  • To further simplify, multiply by \( x^2/x^2 \) to obtain \( \frac{x^2}{x^2 + 1} \).
This expression represents the sum of the series when the condition \( |x| > 1 \) is met.
Common Ratio
The common ratio in a geometric series describes the factor by which we multiply each term to get the next term. It is a crucial part of identifying both the behavior and convergence of the series.
In our given series, the common ratio is \( r = (-1) x^{-2} \).
  • Notice that this ratio involves a combination of the term \((-1)^n\) and \(x^{-2n}\).
  • The sign alternates because of \((-1)^n\), bringing about a change in the series terms from positive to negative or vice versa.
  • The magnitude is controlled by \(x^{-2}\), which is why the series behavior is highly dependent on the value of \(x\).
  • For convergence, only values of \(x\) such that \(|x| > 1\) will make the absolute value of the ratio terms smaller than 1.
Geometric Progression
A geometric progression, or geometric series, is a sequence of numbers where each term after the first is found by multiplying the previous term by the same number, known as the common ratio.
The series given \( \sum_{n=0}^{\infty}(-1)^{n} x^{-2n} \) is indeed a geometric progression.
  • Identify the pattern: each term is related to its predecessor by the factor \((-1)x^{-2}\).
  • Starting with the initial term \( a = 1 \), the series progresses based on the ratio rules.
  • The progression properties allow us to apply specific formulas to assess convergence and compute the series sum.
Understanding geometric progression is essential to solving such series problems because it frames how each term builds upon the last, utilizing the common ratio.

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Most popular questions from this chapter

If the terms of one sequence appear in another sequence in their given order, we call the first sequence a sub-sequence of the second. Prove that if two sub-sequences of a sequence \(\left\\{a_{n}\right\\}\) have different limits \(L_{1} \neq L_{2}\) then \(\left\\{a_{n}\right\\}\) diverges.

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