fn = = 2 n²+n−1 3n2-10

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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For each sequence, determine the limit and prove your claim.

Analysis. Please prove using the epsilon method

The formula depicted in the image is:

\[ f_n = \frac{n^2 + n - 1}{3n^2 - 10} \]

This equation represents a mathematical function \( f_n \) in terms of \( n \). The numerator is \( n^2 + n - 1 \), and the denominator is \( 3n^2 - 10 \). This function can be explored for various values of \( n \) and analyzed to understand its behavior and characteristics.
Transcribed Image Text:The formula depicted in the image is: \[ f_n = \frac{n^2 + n - 1}{3n^2 - 10} \] This equation represents a mathematical function \( f_n \) in terms of \( n \). The numerator is \( n^2 + n - 1 \), and the denominator is \( 3n^2 - 10 \). This function can be explored for various values of \( n \) and analyzed to understand its behavior and characteristics.
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Follow-up Question

Hi! Thank you for helping me! Can you please explain why we claim for all n>=3 here?

Let €70
For
be arbitrany number.
| fn
ам пуз
w/-
Transcribed Image Text:Let €70 For be arbitrany number. | fn ам пуз w/-
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Follow-up Question

Why here is < ?

(3n+7)/(9n^2 - 49) should be smaller than (3n+7)/(9n^2 - 30) right?

The image contains mathematical expressions comparing two rational expressions and performing factorization. Here is the transcription:

1. Starting Expression:
   \[
   \frac{3n + 7}{9n^2 - 30}
   \]

2. Comparison with another expression:
   \[
   < \frac{3n^2 + 7}{9n^2 - 49}
   \]

3. Further simplification of the starting expression:
   \[
   = \frac{3n + 7}{(3n + 7)(3n - 7)}
   \]

The red circle highlights the comparison between the initial expression and the expression to which it is being compared. The inequality symbol "<" indicates that the first expression is less than the second.

Explanation of Simplification:

- The denominator \(9n^2 - 49\) in the second expression is a difference of squares, which can be factored as \((3n + 7)(3n - 7)\). 
- The denominator of the initial expression \(9n^2 - 30\) is not factored in the transcription.

This transcription helps illustrate algebraic manipulation through comparison and simplification of rational expressions.
Transcribed Image Text:The image contains mathematical expressions comparing two rational expressions and performing factorization. Here is the transcription: 1. Starting Expression: \[ \frac{3n + 7}{9n^2 - 30} \] 2. Comparison with another expression: \[ < \frac{3n^2 + 7}{9n^2 - 49} \] 3. Further simplification of the starting expression: \[ = \frac{3n + 7}{(3n + 7)(3n - 7)} \] The red circle highlights the comparison between the initial expression and the expression to which it is being compared. The inequality symbol "<" indicates that the first expression is less than the second. Explanation of Simplification: - The denominator \(9n^2 - 49\) in the second expression is a difference of squares, which can be factored as \((3n + 7)(3n - 7)\). - The denominator of the initial expression \(9n^2 - 30\) is not factored in the transcription. This transcription helps illustrate algebraic manipulation through comparison and simplification of rational expressions.
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