Discussing the Convergence of a Series: Get My Opinion!

[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?
many thanks in advance

 

[pasmith]

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.

 

[Mark44]

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?
many thanks in advance

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]

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!

 

[Amaelle]

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!