Chapter 1: Problem 1

\(\sum_{n=1}^{\infty} \frac{1}{2^{n}}\)

Short Answer

Expert verified

The sum of the series is 1.

Step by step solution

Identify the Series

Recognize that the problem gives an infinite geometric series: \textbf{Series:} \(\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \bigcdots \)

Find the First Term (\(a\))

The first term of the geometric series is \(\frac{1}{2}\).

Determine the Common Ratio (\(r\))

The common ratio between successive terms is \(\frac{1}{2}\), since each term after the first is obtained by multiplying the previous term by \(\frac{1}{2}\).

Use the Infinite Geometric Series Sum Formula

The sum \(S\) of an infinite geometric series with first term \(a\) and common ratio \(r\) (\( |r| < 1 \)) is given by: \[ S = \frac{a}{1 - r} \]

Substitute Values

Substitute \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\) into the formula: \[ S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} \]

Simplify the Expression

Simplify the right-hand side: \[ S = \frac{\frac{1}{2}}{\frac{1}{2}} = 1 \]

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Key Concepts

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

Sum of Series

To understand the sum of an infinite geometric series, let's break it down using an example. Consider the series: \(\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \bigcdots \). Each term is a fraction that gets smaller and smaller. Since this series continues indefinitely, it is an 'infinite geometric series'.

To find the 'sum of the series', we believe that if the terms get very small, the sum can approach a particular number. In this case, the sum of the series simplifies to 1. This fascinating result shows that even an infinitely long series can have a finite sum!

Common Ratio

The 'common ratio' is a key element in a geometric series. It is the constant factor you multiply to get from one term to the next. For the series: \(\frac{1}{2}, \frac{1}{2^2}, \frac{1}{2^3}, \frac{1}{2^4}, \bigcdots \), each term is obtained by multiplying the previous term by \(\frac{1}{2}\).

Mathematically, if you divide any term by its preceding term, you will get \(\frac{1}{2}\). This ratio must be consistent throughout the series. Here, the common ratio \(r\) is \(\frac{1}{2}\). Understanding the common ratio helps in applying the geometric series formula correctly.

Geometric Series Formula

The 'geometric series formula' is essential for finding the sum of an infinite geometric series. It is given by: \[ S = \frac{a}{1 - r} \]

Here:

\(S\) represents the sum of the series.
\(a\) is, the first term of the series.
\(r\) is, the common ratio.

For our series \(\frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \bigcdots \), we substitute \(a = \frac{1}{2}\) and \(r = \frac{1}{2}\) into the formula: \[ S = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1 \] This calculation confirms that the sum of the series is 1. The formula works perfectly as long as \(|r| < 1\).

Convergence Criteria

The 'convergence criteria' are rules that determine whether an infinite series has a finite sum. For a geometric series to converge (i.e., have a finite sum), the absolute value of the common ratio must be less than 1 \( |r| < 1 \). If \( |r| \) is not less than 1, the series will diverge, meaning it does not have a finite sum.

In our example, the common ratio is \(\frac{1}{2}\), which is less than 1. Therefore, the series converges, and we can use the geometric series formula to find its sum. Without meeting this criterion, the formula would not give a valid sum.