**Selecting and Summing Convergent Geometric Series**The goal is to identify the convergent geometric series from the following options and then determine its sum:1. $ \sum_{n=1}^{\infty} \frac{1}{n} $2. $ \sum_{n=1}^{\infty} 2(1.01)^n $3. $ \sum_{n=1}^{\infty} 2(0.99)^{n-1} $**Explanation:**- The first series is the harmonic series, which is known to diverge.- The second series has a common ratio of 1.01, which is greater than 1, making it a divergent geometric series.- The third series has a common ratio of 0.99, which is less than 1, indicating that it is a convergent geometric series.To calculate the sum of the convergent geometric series $ \sum_{n=1}^{\infty} 2(0.99)^{n-1} $, use the formula for the sum of an infinite geometric series:\[ S = \frac{a}{1 - r} \]where $ a = 2 $ is the first term and $ r = 0.99 $ is the common ratio.Thus, the sum is:\[ S = \frac{2}{1 - 0.99} = \frac{2}{0.01} = 200 \]

Given the series: ∑n=1∞1n,∑n=1∞21.01n,∑n=1∞20.99n

Answered: Pick out the conveyent geometcii seies…

Math

Calculus

Pick out the conveyent geometcii seies and Find its sum :

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

Angles

In mathematics, angles constitute a very important part. Just like the concept of lines, an angle is a necessary figure that is used for measurement. How do you define its formation? It is formed when two or more lines meet at a point. The point at which two or more lines meet is known as the vertex. When the lines get inclined at this point, they do form various angles. Angles are measured from the vertex in terms of degrees.

Question

**Selecting and Summing Convergent Geometric Series**

The goal is to identify the convergent geometric series from the following options and then determine its sum:

1. $ \sum_{n=1}^{\infty} \frac{1}{n} $

2. $ \sum_{n=1}^{\infty} 2(1.01)^n $

3. $ \sum_{n=1}^{\infty} 2(0.99)^{n-1} $

**Explanation:**

- The first series is the harmonic series, which is known to diverge.

- The second series has a common ratio of 1.01, which is greater than 1, making it a divergent geometric series.

- The third series has a common ratio of 0.99, which is less than 1, indicating that it is a convergent geometric series.

To calculate the sum of the convergent geometric series $ \sum_{n=1}^{\infty} 2(0.99)^{n-1} $, use the formula for the sum of an infinite geometric series:

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

where $ a = 2 $ is the first term and $ r = 0.99 $ is the common ratio.

Thus, the sum is:

\[ S = \frac{2}{1 - 0.99} = \frac{2}{0.01} = 200 \]

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