(1) Find the summation of the series +++++

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...
icon
Related questions
Question
**Problem Statement:**

(1) Find the summation of the series

\[
\frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots
\]

**Explanation:**

This is an infinite geometric series where the first term \(a\) is \(\frac{1}{3}\) and the common ratio \(r\) is also \(\frac{1}{3}\).

The sum \(S\) of an infinite geometric series can be found using the formula:

\[
S = \frac{a}{1 - r}
\]

where \(|r| < 1\).

Applying the values from the series:

- \(a = \frac{1}{3}\)
- \(r = \frac{1}{3}\)

Thus,

\[
S = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2}
\]

Therefore, the summation of the series is \(\frac{1}{2}\).
Transcribed Image Text:**Problem Statement:** (1) Find the summation of the series \[ \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \] **Explanation:** This is an infinite geometric series where the first term \(a\) is \(\frac{1}{3}\) and the common ratio \(r\) is also \(\frac{1}{3}\). The sum \(S\) of an infinite geometric series can be found using the formula: \[ S = \frac{a}{1 - r} \] where \(|r| < 1\). Applying the values from the series: - \(a = \frac{1}{3}\) - \(r = \frac{1}{3}\) Thus, \[ S = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2} \] Therefore, the summation of the series is \(\frac{1}{2}\).
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning