**Problem Statement:**Find the value of the expression:\[ 1 - \frac{7}{1} + \frac{49}{1} - \cdots + (-1)^{28} \frac{7^{28}}{1^{28}}. \]**Explanation:**This expression is a series where each term follows the pattern:\[ (-1)^n \frac{7^n}{1^n}, \]where $ n $ ranges from 0 to 28. The series alternates between addition and subtraction because of the $(-1)^n$ factor. This results in positive terms when $ n $ is even and negative terms when $ n $ is odd.The series can be expressed as:\[ \sum_{n=0}^{28} (-1)^n \frac{7^n}{1^n}. \]**Graphical Representation:**A visual representation of this series is not provided in the image but would involve plotting the sequence of values determined by each term $ (-1)^n 7^n $ along the x-axis from $ n = 0 $ to $ n = 28 $, with heights representing the value of each term. The sum of these heights would approach the total sum of the series.

Name: **Problem Statement:**Find the value of the expression:\[ 1 - \frac{7
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Description: **Problem Statement:**Find the value of the expression:\[ 1 - \frac{7}{1} + \frac{49}{1} - \cdots + (-1)^{28} \frac{7^{28}}{1^{28}}. \]**Explanation:**This expression is a series where each term follows the pattern:\[ (-1)^n \frac{7^n}{1^n}, \]where

The given series sum is 1-71+491-…+-128728128. The first term of the above series is 1 which can be…

Answered: + - ..+(-1)28 . ) 28_728 15

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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**Problem Statement:**

Find the value of the expression:

\[ 1 - \frac{7}{1} + \frac{49}{1} - \cdots + (-1)^{28} \frac{7^{28}}{1^{28}}. \]

**Explanation:**

This expression is a series where each term follows the pattern:

\[ (-1)^n \frac{7^n}{1^n}, \]

where $ n $ ranges from 0 to 28. The series alternates between addition and subtraction because of the $(-1)^n$ factor. This results in positive terms when $ n $ is even and negative terms when $ n $ is odd.

The series can be expressed as:

\[ \sum_{n=0}^{28} (-1)^n \frac{7^n}{1^n}. \]

**Graphical Representation:**

A visual representation of this series is not provided in the image but would involve plotting the sequence of values determined by each term $ (-1)^n 7^n $ along the x-axis from $ n = 0 $ to $ n = 28 $, with heights representing the value of each term. The sum of these heights would approach the total sum of the series.

Expert Solution

Step 1

The given series sum is $1 - \frac{7}{1} + \frac{49}{1} - \dots + {(- 1)}^{28} \frac{7^{28}}{1^{28}}$ .

The first term of the above series is 1 which can be considered as $1 = 7^{0}$ .

The second term is $- \frac{7}{1} = - 7$ .

The third term is $\frac{49}{1} = 7^{2}$ .

The fourth term would be $- \frac{343}{1} = - 7^{3}$ .

So, in general odd numbered terms have positive sign while even numbered term has negative sign.

The 29th term (odd numbered term) is ${(- 1)}^{28} \frac{7^{28}}{1^{28}} = 7^{28}$ .

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