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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Question
What is the sum of the infinite series?
please show me the process on how this is completed, I am extremely stuck.
(Those are n, not pi) sorry the picture isn't the best
![The following expression appears in the image:
\[ \sum_{n=1}^{\infty} \frac{1}{10^{2n}} \]
This notation signifies the infinite series or sum starting from \( n = 1 \) to infinity, for the term \( \frac{1}{10^{2n}} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff1236809-23f1-4bf3-a287-e517d49ba79e%2F41a17e34-e112-4f37-b48b-c40774e24529%2Fzahru3o.jpeg&w=3840&q=75)
Transcribed Image Text:The following expression appears in the image:
\[ \sum_{n=1}^{\infty} \frac{1}{10^{2n}} \]
This notation signifies the infinite series or sum starting from \( n = 1 \) to infinity, for the term \( \frac{1}{10^{2n}} \).
![The image contains a mathematical expression involving a summation. Below is the transcription suitable for an educational website:
---
**Mathematical Summation Expression:**
The expression shown is:
\[
\sum_{n=1}^{\infty} \frac{1}{10^{n}}
\]
This denotes the sum of the infinite series starting from \( n = 1 \) to \( \infty \) where each term of the series is given by \( \frac{1}{10^{n}} \).
---
**Explanation:**
- **Summation Symbol (Σ):** This symbol represents the sum of a sequence of terms.
- **Limits of Summation:** The \( n=1 \) at the bottom of the summation symbol indicates that the summation starts at \( n = 1 \). The \( \infty \) at the top indicates that the summation goes to infinity.
- **Term of the Series:** The term in the series for each \( n \) is given by \( \frac{1}{10^{n}} \).
**Understanding the Series:**
- For \( n = 1 \), the term is \( \frac{1}{10^{1}} = \frac{1}{10} \).
- For \( n = 2 \), the term is \( \frac{1}{10^{2}} = \frac{1}{100} \).
- For \( n = 3 \), the term is \( \frac{1}{10^{3}} = \frac{1}{1000} \).
And so on, adding each successive term where the denominator is a higher power of 10.
**Series Convergence:**
This geometric series converges, and the sum can be calculated using the formula for the sum of an infinite geometric series:
\[
S = \frac{a}{1 - r}
\]
where \( a \) is the first term and \( r \) is the common ratio. In this case:
\[
a = \frac{1}{10}, \quad r = \frac{1}{10}
\]
So, the sum \( S \) is:
\[
S = \frac{\frac{1}{10}}{1 - \frac{1}{10}} = \frac{\frac{1}{10}}{\frac{9}{10}} = \frac{1}{9}
\]
Thus, the sum of the series](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff1236809-23f1-4bf3-a287-e517d49ba79e%2F41a17e34-e112-4f37-b48b-c40774e24529%2F4ak3xop.jpeg&w=3840&q=75)
Transcribed Image Text:The image contains a mathematical expression involving a summation. Below is the transcription suitable for an educational website:
---
**Mathematical Summation Expression:**
The expression shown is:
\[
\sum_{n=1}^{\infty} \frac{1}{10^{n}}
\]
This denotes the sum of the infinite series starting from \( n = 1 \) to \( \infty \) where each term of the series is given by \( \frac{1}{10^{n}} \).
---
**Explanation:**
- **Summation Symbol (Σ):** This symbol represents the sum of a sequence of terms.
- **Limits of Summation:** The \( n=1 \) at the bottom of the summation symbol indicates that the summation starts at \( n = 1 \). The \( \infty \) at the top indicates that the summation goes to infinity.
- **Term of the Series:** The term in the series for each \( n \) is given by \( \frac{1}{10^{n}} \).
**Understanding the Series:**
- For \( n = 1 \), the term is \( \frac{1}{10^{1}} = \frac{1}{10} \).
- For \( n = 2 \), the term is \( \frac{1}{10^{2}} = \frac{1}{100} \).
- For \( n = 3 \), the term is \( \frac{1}{10^{3}} = \frac{1}{1000} \).
And so on, adding each successive term where the denominator is a higher power of 10.
**Series Convergence:**
This geometric series converges, and the sum can be calculated using the formula for the sum of an infinite geometric series:
\[
S = \frac{a}{1 - r}
\]
where \( a \) is the first term and \( r \) is the common ratio. In this case:
\[
a = \frac{1}{10}, \quad r = \frac{1}{10}
\]
So, the sum \( S \) is:
\[
S = \frac{\frac{1}{10}}{1 - \frac{1}{10}} = \frac{\frac{1}{10}}{\frac{9}{10}} = \frac{1}{9}
\]
Thus, the sum of the series
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