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Chapter 10: Problem 43

Find the sum of each infinite geometric series, if possible. $$\sum_{n=1}^{\infty}\left(-\frac{1}{3}\right)^{n}$$

Short Answer

Expert verified
The sum of the infinite geometric series is \(-\frac{1}{4}\).

Step by step solution

01

Identify the General Term

The series given is \( \sum_{n=1}^{\infty} \left(-\frac{1}{3}\right)^{n} \). The general term of the series can be written as \( a_n = \left(-\frac{1}{3}\right)^{n} \).
02

Determine the First Term and Common Ratio

The first term, \( a_1 \), of the series is \( \left(-\frac{1}{3}\right)^{1} = -\frac{1}{3} \). The common ratio \( r \) is \( -\frac{1}{3} \).
03

Check the Sum Condition

For an infinite geometric series to have a sum, the absolute value of the common ratio \( r \) must be less than 1. Here, \( |r| = \left| -\frac{1}{3} \right| = \frac{1}{3} < 1 \). Hence, the series sum can be calculated.
04

Apply the Sum Formula for Infinite Series

The sum \( S \) of an infinite geometric series with first term \( a_1 \) and common ratio \( r \), where \(|r| < 1\), is given by the formula: \[ S = \frac{a_1}{1 - r} \] Substitute \( a_1 = -\frac{1}{3} \) and \( r = -\frac{1}{3} \) into the formula: \[ S = \frac{-\frac{1}{3}}{1 - (-\frac{1}{3})} = \frac{-\frac{1}{3}}{1 + \frac{1}{3}} \]
05

Simplify the Expression

Calculate the denominator: \[ 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \] Now, simplify the fraction for \( S \): \[ S = \frac{-\frac{1}{3}}{\frac{4}{3}} = -\frac{1}{3} \times \frac{3}{4} = -\frac{1}{4} \]

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Key Concepts

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

Sum of Infinite Series
An infinite series sums an endless list of numbers. In the case of a geometric series, this means summing numbers that follow a constant factor from one term to the next. Even though the series is infinite, it can still add up to a finite number. This magic happens under certain conditions. When the absolute value of the common ratio is less than 1, the values get smaller and smaller, allowing the series to converge to a finite sum.

The formula for the sum of an infinite geometric series is crucial:
  • First term, noted as \(a_1\).
  • Common ratio, \(r\), should satisfy \(|r| < 1\).
  • The sum formula is \(S = \frac{a_1}{1 - r}\).
Given these points, one can calculate the sum smoothly, as shown through our original exercise. When the common ratio's absolute value is less than 1, the series converges, promising a definite sum. For instance, the series \(\sum_{n=1}^{\infty}\left(-\frac{1}{3}\right)^{n}\) converges to \(-\frac{1}{4}\).
Common Ratio
The common ratio is a key element in any geometric series. It dictates how each term relates to the one before it. A consistent ratio helps recognize a series as geometric easily.

In our example, the common ratio \(r\) is \(-\frac{1}{3}\). This means each term is \(-\frac{1}{3}\) times the previous one. The common ratio is important because it tells whether the series converges to a sum. To ensure convergence, \(|r|\) must be less than 1. This constraint limits the growth of the terms, helping them to shrink progressively and allowing a summable result.

A common ratio鈥檚 magnitude spotlights if a series will settle to a finite number or diverge endlessly. Resembling links in a chain, the ratio binds the series, steering it towards addition or expansion.
Geometric Series Formula
The geometric series formula is a mathematical tool used to find the sum of both finite and infinite geometric series. It's tailored for situations where each term is a constant multiple of the previous one. For finite series, it's straightforward; however, infinite series conditions make the formula special.

When dealing with an infinite geometric series, its sum is determined by:
  • First term \(a_1\)
  • Common ratio \(r\) (with \(|r| < 1\))
  • The formula is \(S = \frac{a_1}{1 - r}\)
This formula demands that the common ratio has an absolute value less than 1. Implementing this ensures the series doesn鈥檛 spiral into infinity, but instead converges
to a finite sum.

Applying the geometric series formula requires just a few steps 鈥� identify the first term, note the common ratio, check its magnitude, then plug into the formula. As the formula provides clarity and efficiency, mastering it paves the way for tackling diverse and complex problems involving geometric sequences.

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Most popular questions from this chapter

An attorney is trying to calculate the costs associated with going into private practice. If she hires a paralegal to assist her, she will have to pay the paralegal 20.00 dollars per hour. To be competitive with most firms, she will have to give her paralegal a 2 dollars per hour raise each year. Find a general term of a sequence \(a_{n}\) that would represent the hourly salary of a paralegal with \(n\) years of experience. What will be the paralegal's salary with 20 years of experience?

In calculus, we study the convergence of sequences. A sequence is convergent when its terms approach a limiting value. For example, \(a_{n}=\frac{1}{n}\) is convergent because its terms approach zero. If the terms of a sequence satisfy \(a_{1} \leq a_{2} \leq a_{3} \leq \ldots \leq a_{n} \leq \ldots\) the sequence is monotonic nondecreasing. If \(a_{1} \geq a_{2} \geq a_{3} \geq \ldots \geq a_{n} \geq \ldots,\) the sequence is monotonic nonincreasing. Classify each sequence as monotonic or not monotonic. If the sequence is monotonic, determine whether it is nondecreasing or nonincreasing. $$a_{n}=\sin \left(\frac{n \pi}{4}\right)$$

In calculus, the difference quotient \(\frac{f(x+h)-f(x)}{h}\) of a function \(f\) is used to find the derivative of the function \(f\). Use the Binomial theorem to find the difference quotient of each function. $$f(x)=(2 x)^{n}$$

The future value of an ordinary annuity is given by the formula \(F V=P M T\left[\left((1+i)^{n}-1\right) / i\right]\) where \(P M T=\) amount paid into the account at the end of each period, \(i=\) interest rate per period, and \(n=\) number of compounding periods. If you invest 5,000 dollars at the end of each year for 5 years, you will have an accumulated value of \(F V\) as given in the above formula at the end of the \(n\) th year. Determine how much is in the account at the end of each year for the next 5 years if \(i=0.06\).

If the inflation rate is \(3.5 \%\) per year and the average price of a home is 195,000 dollars, the average price of a home after \(n\) years is given by \(A_{n}=195,000(1.035)^{n} .\) Find the average price of the home after 6 years.
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