I can't seem to understand the ratio test proof

[A.MHF]

Hi everyone, I'm currently taking Calc II course and I'm kind of stuck in this ratio test proof thing.

Homework Statement

http://blogs.ubc.ca/infiniteseriesmodule/appendices/proof-of-the-ratio-test/proof-of-the-ratio-test/

I'm trying to understand the proof, but there are some parts that I don't really get.
So assuming that |a_n+1/a_n| = L < 1
and that L < r
|a_n+1/a_n| < r
but then, the proof says there is an integer N which is < or = to n. From where did that come from? Why do we need it in the first place? And how does it relate to the proof?
The proof goes on and I don't really have any idea what's going on.

Homework Equations

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The Attempt at a Solution

I tried making a number line to understand what's going on but this seems so subtle.

 

[Delta2]

there are some small typos in the proof which maybe make it harder to understand it, but anyway that N comes from the definition of the limit of a sequence (this is stated in the proof that it follows from the formal definition of the limit). Intuitively when we know that a sequence C_n has limit L this means that C_n gets closer to L as n grows larger. So if we want Cn to be close enough to L so that it is L<C_n<r all we have to do is to choose a large enough N and for all n>=N it will be L<C_n<r.

The proof utilizes of this fact for the sequence C_n=|a_n+1/a_n|. It uses this N to prove that |a_N+k|<|a_N|r^k. From this last inequality you can prove that the series $\sum_{k=1}^{\infty}|a_{N+k}|$ converges (because it is bounded by the geometric series $\sum_{k=1}^{\infty}|a_{N}|r^k$ which obviously converges). Now the series $\sum_{k=1}^{\infty}|a_{k}|$ also converges because it differs from the $\sum_{k=1}^{\infty}|a_{N+k}|$ only by a finite sum $\sum_{k=1}^{N}|a_k|$

 

[vela]

Hi everyone, I'm currently taking Calc II course and I'm kind of stuck in this ratio test proof thing.

Homework Statement

http://blogs.ubc.ca/infiniteseriesmodule/appendices/proof-of-the-ratio-test/proof-of-the-ratio-test/

I'm trying to understand the proof, but there are some parts that I don't really get.
So assuming that |a_n+1/a_n| = L < 1

The proof doesn't say $\lvert a_{n+1}/a_n \rvert = L$. It says that
$$\lim_{n \to \infty} \left\lvert \frac{a_{n+1}}{a_n} \right\rvert = L.$$ If you understand what the limit means, the proof should make more sense to you.

 

[A.MHF]

there are some small typos in the proof which maybe make it harder to understand it, but anyway that N comes from the definition of the limit of a sequence (this is stated in the proof that it follows from the formal definition of the limit). Intuitively when we know that a sequence C_n has limit L this means that C_n gets closer to L as n grows larger. So if we want Cn to be close enough to L so that it is L<C_n<r all we have to do is to choose a large enough N and for all n>=N it will be L<C_n<r.

The proof utilizes of this fact for the sequence C_n=|a_n+1/a_n|. It uses this N to prove that |a_N+k|<|a_N|r^k. From this last inequality you can prove that the series $\sum_{k=1}^{\infty}|a_{N+k}|$ converges (because it is bounded by the geometric series $\sum_{k=1}^{\infty}|a_{N}|r^k$ which obviously converges). Now the series $\sum_{k=1}^{\infty}|a_{k}|$ also converges because it differs from the $\sum_{k=1}^{\infty}|a_{N+k}|$ only by a finite sum $\sum_{k=1}^{N}|a_k|$

Thanks, that makes much more sense now!

Vela: yea sorry, I knew about the limit I just forgot to type it.