Limits of Sequences Homework: Proving Limit of a_n/n = 0

In summary, according to the definition of a limit, if the limit of a sequence is 2 as n goes to infinity, then the limit of the sequence divided by n approaches 0 as n goes to infinity. This can be shown by choosing any epsilon greater than 0 and finding an M such that if n is greater than M, the absolute value of the sequence divided by n is less than epsilon.
  • #1
NickMusicMan
7
0

Homework Statement



For a sequence a_n:

If lim (a_n) =2, use the definition of a limit to show that lim (a_n / n) = 0

all limits are as n goes to infinity

The Attempt at a Solution



I know that I need to show:

Give any [tex]\epsilon[/tex]>0 there is some M so that

if n>M then |a_n / n| < [tex]\epsilon[/tex]But I can't Figure out how to do that.

Any help would be much appreciated! I have an exam in a few days and I came across this as a practice problem.

-NN
 
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  • #2
by the way: I totally understand intuitively why this is the case, I just figure out how to express it formally.

I know that as n approaches infinity, the fraction approaches (a fixed number)/(infinity) , which means it approaches 0. How can I write this using the formal definition?
 
  • #3
You know the limit of an, so you know there exists an N such that for n>N, |an-2|<ε.

Can you tweak that to get something that'll bound |an/n| from above? Or maybe use the triangle inequality?
 
  • #4
I have tried using the triangle inequality to do so, but i haven't figured anything out yet.
 
  • #5
NickMusicMan said:
I have tried using the triangle inequality to do so, but i haven't figured anything out yet.

Pick a value of epsilon. Then just divide the formal definition of limit by n.
 

FAQ: Limits of Sequences Homework: Proving Limit of a_n/n = 0

What is the purpose of proving the limit of a sequence a_n/n = 0?

The purpose of proving the limit of a sequence a_n/n = 0 is to show that as the index n of the sequence approaches infinity, the values of the sequence approach 0. This is important in understanding the behavior of sequences and their convergence to a specific value.

How do you prove the limit of a sequence a_n/n = 0?

To prove the limit of a sequence a_n/n = 0, you must show that for any small positive number ε, there exists a natural number N such that whenever n>N, the absolute value of a_n/n is less than ε. This can be done using the definition of a limit, various limit theorems, and algebraic manipulations.

What are some common techniques used to prove the limit of a sequence a_n/n = 0?

Some common techniques used to prove the limit of a sequence a_n/n = 0 include the squeeze theorem, the ratio test, and the comparison test. These techniques allow for a simplified and more manageable approach to proving the limit.

Can the limit of a sequence a_n/n = 0 be proven using only algebraic manipulations?

No, the limit of a sequence a_n/n = 0 cannot be proven using only algebraic manipulations. While algebraic manipulations can help simplify the expression and make it easier to work with, they are not enough to prove the limit. Other techniques, such as the ones mentioned in question 3, are also necessary.

What are the implications of proving the limit of a sequence a_n/n = 0?

Proving the limit of a sequence a_n/n = 0 has several implications. It shows that the sequence is convergent and approaches 0 as the index n approaches infinity. This can be useful in various fields of mathematics, such as calculus and analysis, where the behavior of sequences is important in understanding functions and their limits.

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