# Question about one part of the Ratio Test proof

• ivan77
In summary, Ivan77 is trying to figure out how to use the ratio test to prove that the absolute value of the sum of two consecutive integers is greater than the absolute value of the sum of the first integer plus the absolute value of the second integer. He is confused by the proof because he does not understand the logic behind the inequality given.
ivan77
Hi,

there is a proof of the ratio test that I have seen a couple of times here:

http://tutorial.math.lamar.edu/Classes/CalcII/RatioTest.aspxI don't know how to use latex, so abs(x) will mean absolute value

I am ok with this:

abs( aN+1) < r abs(aN)
I don't understand where the r2 come from in the next line
abs( aN+2) < r abs(aN+1)< r2 abs(aN)

Similarly, I don't understand how to get to the generalization of
abs( aN+k) < rk abs(aN+1)I should probably add that I understand all other aspects (i.e. using comparison test), but since I don't understand this part, my understanding of the proof fails.

Thanks,

Ivan77

Last edited:
Just in case your were wondering, you don't need LaTeX for absolute value, the key is right above your Enter key on your keyboard (PC): |

So we know $\left|\frac{a_{n+1}}{a_n}\right| < r$ which is the same as $|a_{n+1}| < r |a_n|$. This is saying that the n+1st term (the term after the nth term) in the sequence is the nth term multiplied by r.

What they are doing in the proof is going backwards. Think of it this way, it might help you better visualize it:

Starting with $|a_{N+1}| < r |a_N|$ and $|a_{N+2}| < r |a_{N+1}|$ we can substitute for $|a_{N+1}|$ below:

$\begin{eqnarray*} |a_{N+2}| &<& r |a_{N+1}|\\ &<& r (r |a_{N}|) = r^2 |a_N| \end{eqnarray*}$

And you can continue this process

$\begin{eqnarray*} |a_{N+3}| &<& r |a_{N+2}|\\ &<& r (r^2 |a_{N}|) = r^3 |a_N| \end{eqnarray*}$

Does that help or is it still confusing?

Hugely appreciate the reply scurty! It answered my question. I've been staring at this proof for a couple of hours.

I was getting stuck on the logic of how we would know that the r^2 would not make the term less than the An+1 rather that using the inequality given.

This is what I was gettign stuck on:

r|aN+1| < r(r|aN|)

has to be true since

since
|aN+1| < r|aN|

and multiplying by a factor of r will not change that fact.

Why couldn't I see that simply distributing the r in the right term would answer the question?
The odd thing about my self studying calculus is that the concepts as well answering questions/application is not a problem, but now and then, I get totally stuck on what usually ends up being a minor point.

Any tips?

## 1. What is the Ratio Test?

The Ratio Test is a convergence test used in mathematics to determine whether an infinite series converges or diverges. It is based on the comparison of the ratio of consecutive terms in the series to a limit value.

## 2. How is the Ratio Test used in proof?

In the Ratio Test proof, the test is used to show that if the limit of the ratio of consecutive terms is less than 1, then the series converges. This is a key step in proving the convergence of a series using the Ratio Test.

## 3. What is the limit comparison test?

The limit comparison test is another convergence test that can be used to determine the convergence or divergence of a series. It involves comparing the given series to a known series with a known convergence or divergence, and using their ratio to determine the convergence or divergence of the given series.

## 4. Is the Ratio Test always applicable?

No, the Ratio Test is not always applicable. It can only be used for series whose terms are positive and decreasing. If the terms are not positive and decreasing, then other convergence tests must be used.

## 5. How do I know when to use the Ratio Test in a proof?

The Ratio Test is typically used when the terms of the series involve powers of n, exponentials, or factorials. It is also useful when the terms are complicated and difficult to work with. However, it is always important to consider other convergence tests as well before choosing to use the Ratio Test.

• Calculus
Replies
6
Views
1K
• Calculus
Replies
1
Views
1K
• Calculus
Replies
4
Views
2K
• Calculus
Replies
2
Views
2K
• Calculus
Replies
3
Views
487
• Calculus
Replies
29
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
5
Views
3K
• Calculus
Replies
8
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
994
• Calculus
Replies
11
Views
2K