Does Converging Sequence s_n Imply Converging Average t_n?

[philosophking]

Analysis problem (sequences)--please help

Here is the definition:

t_n = [s_1 + s_2 + ... + s_n] / n ; n >/= 1

I have to show that if lim n-> [infinity] s_n = s, then lim n-> [infinity] t_n = s

First of all, I don't think it's true. Because if s is finite, then lim s/n as n-> [infinity] would be zero, right? And thus lim t_n as n-> [infinity] is zero, and they're not the same.

I'm just wondering how to go about this problem. Thank you.

 

[Hurkyl]

I think I would try it directly with the epsilon-delta definition of a limit.


BTW, you should be able to convince yourself that the limit of t_n is not always zero by considering a simple example.

 

[wais]

The analysis problem you have presented deals with sequences, specifically the limit of a sequence. In order to solve this problem, we need to understand the definitions and properties of limits and sequences.

A sequence is a list of numbers arranged in a specific order. In this case, the sequence is denoted by t_n and is defined as the average of the first n terms of another sequence, denoted by s_n. The average is calculated by adding all the terms and dividing by the number of terms.

A limit is the value that a sequence approaches as the number of terms increases. In this case, we are interested in the limit as n approaches infinity. This means that we are looking at the behavior of the sequence as the number of terms becomes larger and larger.

Now, let's look at the first statement: if lim n-> [infinity] s_n = s. This means that as n approaches infinity, the terms of the sequence s_n get closer and closer to the value s. In other words, the terms of the sequence s_n are approaching the limit s.

Next, we have to show that if this statement is true, then lim n-> [infinity] t_n = s. This means that if the terms of the sequence s_n are approaching the limit s, then the terms of the sequence t_n, which are the averages of the terms of s_n, are also approaching the limit s.

To prove this, we can use the definition of the limit. We have to show that for any small positive number ε, there exists a positive integer N such that for all n > N, the difference between t_n and s is less than ε. In other words, we have to show that t_n is getting closer and closer to s as n approaches infinity.

Since we know that lim n-> [infinity] s_n = s, we can choose a large enough N such that for all n > N, the difference between s_n and s is less than ε/2. This means that for all n > N, we have:

|s_n - s| < ε/2

Now, let's look at the definition of t_n:

t_n = [s_1 + s_2 + ... + s_n] / n

We can rewrite this as:

t_n = [(s_1 - s) + (s_2 - s) + ... + (s_n - s) + ns] / n

Notice that