I need to know why this is correct ### Sums of an Infinite SeriesFind 10 partial sums of the series. (Round your answers to five decimal places.)\[\sum_{n=1}^{\infty} \frac{4}{(-3)^n}\]#### Table of Partial Sums| $ n $ | $ s_n $ ||--------|--------------|| 1 | -1.33333 || 2 | -0.88889 || 3 | -1.03704 || 4 | -0.98765 || 5 | -1.00412 || 6 | -0.99863 || 7 | -1.00046 || 8 | -0.99986 || 9 | -1.00005 || 10 | -0.99999 |#### Graphical Representation**Graph both the sequence of terms and the sequence of partial sums on the same screen.****Graph Explanation:**- **X-Axis**: Represents the number of terms $ n $.- **Y-Axis**: Represents the values of the terms and partial sums.In the graphs provided:- The **black points** represent the sequence of individual terms $\frac{4}{(-3)^n}$.- The **blue points** represent the sequence of partial sums $s_n$.**Key Observations:**- The sequence of partial sums (blue points) oscillates and converges towards a limit.- The individual term values (black points) alternate and decrease in magnitude, gradually approaching zero.These graphs visually illustrate how the series converges as the number of terms increases. In this educational content, we explore the convergence or divergence of a series using graphical representations. Four graphs are presented to assist in understanding the behavior of the series over a domain.### Graphs Description1. **Top-left Graph** - **Axes**: The horizontal axis (x) ranges from 0 to 10, and the vertical axis (y) ranges from -1.0 to 1.0. - **Data Points**: This graph shows black and blue points. The blue points approach zero as x increases.2. **Top-right Graph** - **Axes**: The horizontal axis (x) ranges from 0 to 10, and the vertical axis (y) ranges from -1.5 to 1.0. - **Data Points**: This graph also shows black and blue points. The blue points show descent towards zero, indicating a trend towards convergence.3. **Bottom-left Graph** - **Axes**: The horizontal axis (x) ranges from 0 to 10, and the vertical axis (y) ranges from -0.5 to 1.0. - **Data Points**: Similar to the previous graphs, there are black and blue points. This graph exemplifies the approach of the blue points towards zero, reinforcing the concept of convergence.4. **Bottom-right Graph** - **Axes**: The horizontal axis (x) ranges from 0 to 10, and the vertical axis (y) ranges from -1.5 to 0.5. - **Data Points**: This graph follows the same black and blue point pattern. Again, the blue points approach zero, indicating convergence.### Series AnalysisFollowing the analysis of these graphs, we reach the following conclusion:- **Series Convergence**: The series is convergent. - This conclusion is marked by selecting the option "convergent" from a choice of "convergent" or "divergent," which is indicated in the image with a blue circle and a check mark.### Further Instructions- **If the series is convergent, find the sum**: If the series diverges, enter "DIVERGES." This instructional data helps learners understand the evaluation of a series' convergence by utilizing graphical data points effectively.

Name: I need to know why this is correct ### Sums of an Infinite SeriesFind
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Description: I need to know why this is correct ### Sums of an Infinite SeriesFind 10 partial sums of the series. (Round your answers to five decimal places.)\[\sum_{n=1}^{\infty} \frac{4}{(-3)^n}\]#### Table of Partial Sums| $ n $ | $ s_n $ ||--------|---

nth-Partial sum means we have to add n-terms. For example: By following this process and using…

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Find 10 partial sums of the series. (Round your answers to five decimal places.) Σ (-3)" 00 4 n = 1 1 -1.3333: 2 -0.8888! -1.03704 4 -0.98766 |-1.0041: 6 -0.99864 7 |-1.0004: -0.99986 9 -1.0000€ 10 -0.9999! Graph both the sequence of terms and the sequence of partial sums on the same screen. y y 1.0E 1.0 0.5

Find 10 partial sums of the series. (Round your answers to five decimal places.) Σ (-3)" 00 4 n = 1 1 -1.3333: 2 -0.8888! -1.03704 4 -0.98766 |-1.0041: 6 -0.99864 7 |-1.0004: -0.99986 9 -1.0000€ 10 -0.9999! Graph both the sequence of terms and the sequence of partial sums on the same screen. y y 1.0E 1.0 0.5

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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I need to know why this is correct

### Sums of an Infinite Series

Find 10 partial sums of the series. (Round your answers to five decimal places.)

\[
\sum_{n=1}^{\infty} \frac{4}{(-3)^n}
\]

#### Table of Partial Sums

| $ n $ | $ s_n $ |
|--------|--------------|
| 1 | -1.33333 |
| 2 | -0.88889 |
| 3 | -1.03704 |
| 4 | -0.98765 |
| 5 | -1.00412 |
| 6 | -0.99863 |
| 7 | -1.00046 |
| 8 | -0.99986 |
| 9 | -1.00005 |
| 10 | -0.99999 |

#### Graphical Representation

**Graph both the sequence of terms and the sequence of partial sums on the same screen.**

**Graph Explanation:**
- **X-Axis**: Represents the number of terms $ n $.
- **Y-Axis**: Represents the values of the terms and partial sums.

In the graphs provided:

- The **black points** represent the sequence of individual terms $\frac{4}{(-3)^n}$.
- The **blue points** represent the sequence of partial sums $s_n$.

**Key Observations:**
- The sequence of partial sums (blue points) oscillates and converges towards a limit.
- The individual term values (black points) alternate and decrease in magnitude, gradually approaching zero.

These graphs visually illustrate how the series converges as the number of terms increases.

In this educational content, we explore the convergence or divergence of a series using graphical representations. Four graphs are presented to assist in understanding the behavior of the series over a domain.

### Graphs Description

1. **Top-left Graph**
- **Axes**: The horizontal axis (x) ranges from 0 to 10, and the vertical axis (y) ranges from -1.0 to 1.0.
- **Data Points**: This graph shows black and blue points. The blue points approach zero as x increases.

2. **Top-right Graph**
- **Axes**: The horizontal axis (x) ranges from 0 to 10, and the vertical axis (y) ranges from -1.5 to 1.0.
- **Data Points**: This graph also shows black and blue points. The blue points show descent towards zero, indicating a trend towards convergence.

3. **Bottom-left Graph**
- **Axes**: The horizontal axis (x) ranges from 0 to 10, and the vertical axis (y) ranges from -0.5 to 1.0.
- **Data Points**: Similar to the previous graphs, there are black and blue points. This graph exemplifies the approach of the blue points towards zero, reinforcing the concept of convergence.

4. **Bottom-right Graph**
- **Axes**: The horizontal axis (x) ranges from 0 to 10, and the vertical axis (y) ranges from -1.5 to 0.5.
- **Data Points**: This graph follows the same black and blue point pattern. Again, the blue points approach zero, indicating convergence.

### Series Analysis

Following the analysis of these graphs, we reach the following conclusion:

- **Series Convergence**: The series is convergent.
- This conclusion is marked by selecting the option "convergent" from a choice of "convergent" or "divergent," which is indicated in the image with a blue circle and a check mark.

### Further Instructions

- **If the series is convergent, find the sum**: If the series diverges, enter "DIVERGES."

This instructional data helps learners understand the evaluation of a series' convergence by utilizing graphical data points effectively.

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