### Problem Statement from the Image:**Write the first five terms of an infinite geometric series that converges to 2.**### Explanation for Educational Website:This is a mathematical problem that requires identifying the first five terms of an infinite geometric series, which is a sequence where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The series must also converge to a limit of 2.**Key Concepts:**1. **Infinite Geometric Series:** - A series of the form $ a + ar + ar^2 + ar^3 + \cdots $, where $ a $ is the first term and $ r $ is the common ratio. 2. **Convergence of Geometric Series:** - An infinite geometric series $ a + ar + ar^2 + ar^3 + \cdots $ converges if the absolute value of the common ratio $ |r| < 1 $. - The sum $ S $ of an infinite geometric series is given by $ S = \frac{a}{1 - r} $ where $ |r| < 1 $.### Solving the Problem:Given that the series converges to 2, we use the formula for the sum of an infinite geometric series to set up the equation:\[ S = \frac{a}{1 - r} = 2 \]Let's find an example:1. Choose the first term $ a $ and a common ratio $ r $ such that $ |r| < 1 $.2. Solve for the values of $ a $ and $ r $ under the condition that $ \frac{a}{1 - r} = 2 $.One potential solution is:- Let $ a = 1 $ and $ r = \frac{1}{2} $.Then the series is:\[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots \]**First five terms of the series:**\[ 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} \]This series converges to 2 because:\[ \frac{1}{1 - \frac{1}{2

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Math

Calculus

Write the first five terms of an infinite geometric series that converges to 2.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Question

### Problem Statement from the Image:

**Write the first five terms of an infinite geometric series that converges to 2.**

### Explanation for Educational Website:

This is a mathematical problem that requires identifying the first five terms of an infinite geometric series, which is a sequence where each term is found by multiplying the previous one by a fixed, non-zero number called the common ratio. The series must also converge to a limit of 2.

**Key Concepts:**

1. **Infinite Geometric Series:**
- A series of the form $ a + ar + ar^2 + ar^3 + \cdots $, where $ a $ is the first term and $ r $ is the common ratio.

2. **Convergence of Geometric Series:**
- An infinite geometric series $ a + ar + ar^2 + ar^3 + \cdots $ converges if the absolute value of the common ratio $ |r| < 1 $.
- The sum $ S $ of an infinite geometric series is given by $ S = \frac{a}{1 - r} $ where $ |r| < 1 $.

### Solving the Problem:

Given that the series converges to 2, we use the formula for the sum of an infinite geometric series to set up the equation:
\[ S = \frac{a}{1 - r} = 2 \]

Let's find an example:

1. Choose the first term $ a $ and a common ratio $ r $ such that $ |r| < 1 $.
2. Solve for the values of $ a $ and $ r $ under the condition that $ \frac{a}{1 - r} = 2 $.

One potential solution is:
- Let $ a = 1 $ and $ r = \frac{1}{2} $.

Then the series is:
\[ 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots \]

**First five terms of the series:**
\[ 1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16} \]

This series converges to 2 because:
\[ \frac{1}{1 - \frac{1}{2

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