# 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.
1. The problem statement, all variables and given/known data

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 |an+1/an| = L < 1
and that L < r
|an+1/an| < 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.
2. Relevant equations
-
3. The attempt at a solution
I tried making a number line to understand what's going on but this seems so subtle.

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#### Delta2

Homework Helper
Gold Member
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 Cn has limit L this means that Cn gets closer to L as n grows larger. So if we want Cn to be close enough to L so that it is L<Cn<r all we have to do is to choose a large enough N and for all n>=N it will be L<Cn<r.

The proof utilizes of this fact for the sequence Cn=|an+1/an|. It uses this N to prove that |aN+k|<|aN|rk. 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 $\sum_{k=1}^{N}|a_k|$

Last edited:
• A.MHF

#### vela

Staff Emeritus
Homework Helper
Hi everyone, I'm currently taking Calc II course and I'm kind of stuck in this ratio test proof thing.
1. The problem statement, all variables and given/known data

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 |an+1/an| = 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 Cn has limit L this means that Cn gets closer to L as n grows larger. So if we want Cn to be close enough to L so that it is L<Cn<r all we have to do is to choose a large enough N and for all n>=N it will be L<Cn<r.

The proof utilizes of this fact for the sequence Cn=|an+1/an|. It uses this N to prove that |aN+k|<|aN|rk. 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.

"I can't seem to understand the ratio test proof"

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