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Chapter 11: Problem 2

Determine the sums of the following geometric series when they are convergent. $$1+\frac{3}{4}+\left(\frac{3}{4}\right)^{2}+\left(\frac{3}{4}\right)^{3}+\left(\frac{3}{4}\right)^{4}+\cdots$$

Short Answer

Expert verified
The sum of the series is 4.

Step by step solution

01

Identify the common ratio

In a geometric series, each term after the first is found by multiplying the previous term by a constant called the common ratio. Here, the first term is 1, and the second term is \(\frac{3}{4}\). Thus, the common ratio \(r\) is \(\frac{3}{4}\).
02

Check if the series is convergent

A geometric series converges if the absolute value of the common ratio is less than 1. Here, the common ratio \( \frac{3}{4} \) satisfies \( \bigg|\frac{3}{4}\bigg| < 1 \), so the series converges.
03

Use the sum formula for a convergent geometric series

The sum of an infinite convergent geometric series with first term \( a \) and common ratio \( r \) is given by the formula \[ S = \frac{a}{1-r} \]. In this series, \( a = 1 \) and \( r = \frac{3}{4} \).
04

Calculate the sum

Substitute \( a = 1 \) and \( r = \frac{3}{4} \) into the sum formula: \[ S = \frac{1}{1 - \frac{3}{4}} = \frac{1}{\frac{1}{4}} = 4 \]. Thus, the sum of the series is 4.

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Key Concepts

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

Series Convergence
In mathematics, convergence of a series means that as you add more and more terms of the series, you start to approach a fixed value. This fixed value is known as the sum of the series. For a geometric series, convergence depends on the common ratio, denoted by \(r\). A geometric series will only converge if the absolute value of the common ratio is less than 1. Mathematically, this is written as \( |r| < 1 \).

In our exercise, the series given is:
  • 1+
  • \(\frac{3}{4}\)+
  • \(\left(\frac{3}{4}\right)^2\)+
  • \(\left(\frac{3}{4}\right)^3\)+
  • 袄(袄濒别蹿迟(袄蹿谤补肠调3皑调4皑袄谤颈驳丑迟)镑4袄)+鈰�
Here, the common ratio \(r\) is \(\frac{3}{4}\). Because \(\left| r \right| = \left| \frac{3}{4} \right| = 0.75 < 1\), we can conclude that the series converges.
Common Ratio
In a geometric series, the common ratio is the factor by which we multiply each term to get the next term. It is a constant value throughout the series. The common ratio is a crucial aspect because it dictates the behavior of the series.

For example, if our first term is \( a \) and our common ratio is \( r \), the series will look like this:
  • \(a\)
  • \(ar\)
  • \(ar^2\)
  • \(ar^3\)
  • \(ar^4\)
  • ...
To find the common ratio in our given series, we look at the ratio of the second term to the first term:
  • First term: 1
  • Second term: \(\frac{3}{4}\)
  • Common ratio: \( \frac{\frac{3}{4}}{1} = \frac{3}{4} \)


This common ratio is then used to determine whether the series converges and to find the sum if it does.
Sum Formula for Infinite Series
Once we establish that a geometric series is convergent, we can use a specific formula to find its sum. For an infinite geometric series with a common ratio \( r \) (where \( |r| < 1 \)) and first term \( a \), the sum \( S \) is given by: \[ S = \frac{a}{1-r} \]

To apply this formula to our exercise, we first identify the terms:
  • First term (\( a \)): 1
  • Common ratio (\( r \)): \( \frac{3}{4} \)


Now, we substitute these values into the sum formula: \[ S = \frac{1}{1 - \frac{3}{4}} = \frac{1}{\frac{1}{4}} = 4 \] Therefore, the sum of the given infinite geometric series is 4. This result tells us that as we add more and more terms of the series, the total sum will get closer and closer to 4.

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Most popular questions from this chapter

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