Proving the Limit of a Sequence: Any Ideas?

In summary: Here is the code:var quote = document.getElementById("quote");quote.innerHTML = "In summary, you need to find a sufficiently large number N so that an is within e of a, for any e > 0. If you have infinitely many numbers, and "most" of them are close to a, then you can choose a finite number of them so that even of some of the ones you choose are very far from a, so many of them are close to a that the average is close to a.";
  • #1
r4nd0m
96
1
I am really stuck with this excercise:
Let [tex]a_n and b_n[/tex] be two sequences, where [tex]b_n = \frac{a_1 + a_2 + ... + a_n}{n}[/tex] and [tex]lim_(n \rightarrow \infinity) a_n = a[/tex] prove that [tex]lim_(n \rightarrow \infinity) b_n = a[/tex].

I tried to use the definition - [tex]\mid b_n - a \mid < \frac{\mid a_1 - a \mid + \mid a_2 -a \mid + ... + \mid a_n - a \mid}{n}[/tex], but i don't know how to proceed. Any ideas?
 
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  • #2
Okay, try to think about what's going on. You have a sequence an going to a. So for any e > 0, you can choose a sufficiently large N such that an is within e of a, if n > N. So eventually, everything in the sequence gets as close as you want to a. Now bn is the sequence of averages. You want to show that as a sequence approaches a, so does the corresponding sequence of averages. So you want to show that for any e' > 0, you can choose N' sufficiently large so that the average bn is within e of a, for n > N'. Just choose your N' so large that regardless of the ai that are far away from a, there are enough ai, with i < N' close to a so that the average from a1 to aN' is still close to a. If you have infinitely many numbers, and "most" of them are close to a, then you can choose a finite number of them so that even of some of the ones you choose are very far from a, so many of them are close to a that the average is close to a. And of course, if the average is close for N', then it will work for all bn where n > N'. Do you get the idea? If so, then you just need to state it rigorously, and if you get the idea then it shouldn't be hard.
 
  • #3
r4nd0m said:
I am really stuck with this excercise:
Let [tex]a_n \mbox{ and } b_n[/tex] be two sequences, where [tex]b_n = \frac{a_1 + a_2 + \cdots + a_n}{n}[/tex] and [tex]lim_{n \rightarrow \infty} a_n = a[/tex] prove that [tex]lim_{n \rightarrow \infty} b_n = a[/tex].

I tried to use the definition - [tex]\mid b_n - a \mid < \frac{\mid a_1 - a \mid + \mid a_2 -a \mid + ... + \mid a_n - a \mid}{n}[/tex], but i don't know how to proceed. Any ideas?

It didn't display when I loaded it, so I fixed it in quote.
 
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What is a limit of a sequence?

A limit of a sequence is the value that the terms of the sequence approach as the index of the terms increases towards infinity. It is a fundamental concept in calculus and is used to define the behavior of functions at certain points.

How is the limit of a sequence different from the limit of a function?

The limit of a sequence and the limit of a function are two different concepts. The limit of a sequence is defined for a sequence of numbers, while the limit of a function is defined for a function that maps a set of inputs to a set of outputs. Additionally, the limit of a function can be evaluated at a specific point, while the limit of a sequence is evaluated as the index of the terms approaches infinity.

What is a proof of a limit of a sequence?

A proof of a limit of a sequence is a mathematical demonstration that shows the existence of the limit and its value. It involves using the definition of a limit and various mathematical techniques, such as the epsilon-delta method, to show that the terms of the sequence approach a specific value as the index of the terms increases towards infinity.

Why is proving the limit of a sequence important?

Proving the limit of a sequence is important because it allows us to understand the behavior of a sequence and make predictions about its values. It also has practical applications in fields such as engineering, physics, and economics, where the behavior of systems can be described using sequences.

What are some common techniques used in limit of a sequence proofs?

Some common techniques used in limit of a sequence proofs include the squeeze theorem, the monotone convergence theorem, and the Cauchy criterion. These techniques involve manipulating the terms of the sequence or comparing it to other known sequences to show the existence and value of the limit.

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