# Discussing the Convergence of a Series: Get My Opinion!

• Amaelle
In summary, the conversation was about studying the convergence of a series and the speaker's approach to it. They mentioned using a geometric series to determine convergence, but the other person pointed out that it is not applicable in this case. Instead, they suggested using logarithms and L'Hopital's Rule to determine the limit and conclude the divergence of the series.
Amaelle
Homework Statement
studying the convergence of a serie (look at the image)
Relevant Equations
geometric serie, convergence
Good day
I want to study the connvergence of this serie

I already have the solution but I want to discuss my approach and get your opinion about it
it s clear that n^2+5n+7>n^2+3n+1 so 0<(n^2+3n+1)/(n^2+5n+7)<1 so we can consider this as a geometric serie that converge?

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You can't compare to a geometric series; you have a function of $n$ raised to a power which depends on $n$. That is similar to $$\left(1 - \frac 1n\right)^{n} \to e^{-1} > 0.$$ For that reason $\sum_{n=1}^\infty \left(1 - \frac1n\right)^n$ does not converge.

I would write $$\frac{n^2 + 3n + 1}{n^2 + 5n +7} = 1 - \frac{2n + 6}{n^2 + 5n +7}$$ and take logs.

Amaelle
Amaelle said:
Homework Statement:: studying the convergence of a serie (look at the image)
Relevant Equations:: geometric serie, convergence

Good day
I want to study the connvergence of this serie

View attachment 277247
I already have the solution but I want to discuss my approach and get your opinion about it
it s clear that n^2+5n+7>n^2+3n+1 so 0<(n^2+3n+1)/(n^2+5n+7)<1 so we can consider this as a geometric serie that converge?
If you can establish the fact that ##\lim_{n \to \infty}\left( \frac{n^2 + 3n + 1}{n^2 + 5n + 7}\right)^{n^2} \ne 0##, then you can conclude that the series diverges. This limit has the indeterminate form ##[1^\infty]##, so the best way of determining the limit is by the use of logarithms, and getting it to a form in which L'Hopital's Rule can be applied.

Amaelle
pasmith said:
You can't compare to a geometric series; you have a function of $n$ raised to a power which depends on $n$. That is similar to $$\left(1 - \frac 1n\right)^{n} \to e^{-1} > 0.$$ For that reason $\sum_{n=1}^\infty \left(1 - \frac1n\right)^n$ does not converge.

I would write $$\frac{n^2 + 3n + 1}{n^2 + 5n +7} = 1 - \frac{2n + 6}{n^2 + 5n +7}$$ and take logs.
thanks you so much!

Mark44 said:
If you can establish the fact that ##\lim_{n \to \infty}\left( \frac{n^2 + 3n + 1}{n^2 + 5n + 7}\right)^{n^2} \ne 0##, then you can conclude that the series diverges. This limit has the indeterminate form ##[1^\infty]##, so the best way of determining the limit is by the use of logarithms, and getting it to a form in which L'Hopital's Rule can be applied.
thanks so much!

## What is the convergence of a series?

The convergence of a series refers to whether or not the terms of the series approach a finite limit as the number of terms increases. In other words, it determines if the series will eventually come to a definite value or if it will continue to increase without bound.

## What is the importance of discussing the convergence of a series?

Discussing the convergence of a series is important because it helps us understand the behavior of a series and whether or not it can be summed to a finite value. This information is crucial in many areas of mathematics and science, such as in calculus, statistics, and physics.

## What are the different types of convergence for a series?

There are several types of convergence for a series, including absolute convergence, conditional convergence, and divergence. Absolute convergence occurs when the series converges regardless of the order in which the terms are added. Conditional convergence occurs when the series converges only if the terms are added in a specific order. Divergence occurs when the series does not converge to a finite value.

## How is the convergence of a series determined?

The convergence of a series can be determined through various tests, such as the ratio test, the root test, and the integral test. These tests involve analyzing the behavior of the terms in the series and determining if they approach a finite limit or not.

## Why is it important to get an expert opinion when discussing the convergence of a series?

Getting an expert opinion on the convergence of a series is important because it can be a complex and challenging topic. An expert can provide a thorough analysis and understanding of the series, as well as offer valuable insights and strategies for determining its convergence. This can help ensure accurate and reliable results in mathematical and scientific applications.

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