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Chapter 10: Problem 33

Determine whether each infinite geometric series converges or diverges. If it converges, find its sum. $$ \sum_{j=0}^{\infty}\left(-\frac{1}{2}\right)^{j} $$

Short Answer

Expert verified
The series converges with a sum of \( \frac{2}{3} \).

Step by step solution

01

Identify the First Term (a)

The series given is \( \sum_{j=0}^{\infty} \left(-\frac{1}{2}\right)^{j} \). The first term \( a \) is when \( j = 0 \), which is \( \left(-\frac{1}{2}\right)^{0} = 1 \).
02

Determine the Common Ratio (r)

The common ratio \( r \) is obtained by dividing the second term of the series by the first term. Here, the first term is 1 (\( \left(-\frac{1}{2}\right)^0 \)) and the second term is \( -\frac{1}{2} \). Thus, \( r = -\frac{1}{2} \).
03

Check the Condition for Convergence

An infinite geometric series will converge if the absolute value of the common ratio \( |r| < 1 \). Here, \( r = -\frac{1}{2} \), so \( |r| = \frac{1}{2} < 1 \). Therefore, the series converges.
04

Calculate the Sum of the Convergent Series

The sum \( S \) of an infinite convergent geometric series is calculated using the formula \( S = \frac{a}{1-r} \). Substituting the values, \( a = 1 \) and \( r = -\frac{1}{2} \), we get \( S = \frac{1}{1 - (-\frac{1}{2})} = \frac{1}{1 + \frac{1}{2}} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \).

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Key Concepts

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

Infinite Series
An infinite series is a sum of infinitely many terms. It occurs when we try to add up all the terms of an infinite sequence. The series is represented by the sigma symbol \[ \sum_{j=0}^{\infty} a_j \]where \( a_j \) denotes the terms in the sequence. In an infinite geometric series, specifically, the terms can be expressed in the form of \( ar^j \).
  • This is where \( a \) represents the first term.
  • The \( r \) stands for the common ratio between successive terms.
An infinite series may converge or diverge. If it converges, it approaches a finite number as the number of terms approaches infinity. If it diverges, the sum grows without bound.
Common Ratio
The common ratio \( r \) is a crucial factor in determining the behavior of a geometric series. It is obtained by dividing any term in the sequence by its preceding term.For instance, if we consider a geometric sequence:
  • First term \( a = 1 \)
  • Second term \( = -\frac{1}{2} \)
The common ratio is calculated as:\[ r = \frac{-\frac{1}{2}}{1} = -\frac{1}{2} \]This ratio determines whether the series converges or diverges. For convergence, the absolute value of the common ratio \( |r| \) must be less than 1. When \( |r| < 1 \), the terms of the series get smaller, allowing the series to add up to a finite sum.
Sum of Series
Finding the sum of an infinite geometric series when it converges is straightforward with the right formula. If the series converges, the sum \( S \) is given by:\[ S = \frac{a}{1 - r} \]Let's understand this using the series example:
  • First term \( a = 1 \)
  • Common ratio \( r = -\frac{1}{2} \)
Plugging these into the formula,\[ S = \frac{1}{1 - (-\frac{1}{2})} = \frac{1}{1 + \frac{1}{2}} = \frac{1}{\frac{3}{2}} = \frac{2}{3} \]Therefore, the infinite series converges to a sum of \( \frac{2}{3} \), meaning that, as you add more terms, their cumulative sum approaches \( \frac{2}{3} \).
Geometric Sequence
A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio \( r \).In our case, the geometric sequence is:
  • First term (\( a = 1 \))
  • Second term \( (-\frac{1}{2}) \) times the first
  • Third term \( (-\frac{1}{2})^2 \) and so on...
The terms follow the pattern \( 1, -\frac{1}{2}, \frac{1}{4}, -\frac{1}{8}, \ldots \)
  • Notice how each term is the previous term times \(-\frac{1}{2}\).
  • This makes the common ratio \( r = -\frac{1}{2} \).
A geometric sequence provides the foundational structure for a geometric series, helping us understand how the sum of such series behaves, especially when it has infinitely many terms.

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Most popular questions from this chapter

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