Cesaro Sum. Understanding the sequence.

• fatcrispy
In summary, the Cesaro sum of a finite sequence is a new sequence that gives the cumulative average of the given sequence. It can be used in the study of divergent sequences. In the given problem, the Cesaro sum of the 99th term sequence is 2000. Adding a 1 to the beginning of the sequence makes it a 100 term sequence. So the Cesaro sum of the 100th term sequence is also 2000.
fatcrispy

Homework Statement

I have a finite sequence Z=(z1,...,zn). The Cesaro sum of Z is $$\frac{(B1+B2+...+Bn)}{n}$$

BC=z1+z2+...zC (1$$\leq$$C$$\leq$$n)

Lets say the problem asks "The Cesaro sum of the 99th term sequence of (z1,...,z99) is 2000, what is the Cesaro sum of the 100 term sequence (1, z1,...,z99)?

The Attempt at a Solution

I read about Cesaro sum on wikipedia but it didn't elaborate much. Here is where I'm at:

2000=$$\frac{(B1+B2+...+B99)}{99}$$

But, honestly, I have no idea how to solve this because I can't find any info on it.

Call z1 + z2 +...+z99 = S

You are given that S/99 = 2000.

Now you are asked to calculate (1 + S)/100. It isn't that tough...

LCKurtz said:
Call z1 + z2 +...+z99 = S

You are given that S/99 = 2000.

Now you are asked to calculate (1 + S)/100. It isn't that tough...

So, putting 1 in front of the sequence allows z99 to become the 100th term? But it isn't the 100th nth term right? With what you said it would just be 1980.01 as the answer? I am trying to fundamentally understand this. I understand it's just an average but when they start saying the value of C could be lesser or equal to n and all of that I lose the concept.

fatcrispy said:
So, putting 1 in front of the sequence allows z99 to become the 100th term? But it isn't the 100th nth term right? With what you said it would just be 1980.01 as the answer?

Yes, that is the correct answer. Add a 1 to the sequence means there are now 101 terms to average.

I am trying to fundamentally understand this. I understand it's just an average but when they start saying the value of C could be lesser or equal to n and all of that I lose the concept.

The Cesaro sum of a sequence gives you a new sequence which gives cumulative average of the given sequence. One place they are used is in the study of divergent sequences. For example, the sequence 1, -1, 1, -1, 1,... diverges. But, informally, you might say its "average value" is 0. And that is exactly what the Cesaro sum sequence converges to.

LCKurtz said:
Yes, that is the correct answer. Add a 1 to the sequence means there are now 101 terms to average.

Big "oh..." moment. I think I misread the problem. It asks "what is the Cesaro sum of the 100th term sequence (1, z1, ..., z99)?" I realize now that it is the whole 100 term sequence and I don't actually have to go up to z100 right? Because the 1 in front makes it 100 terms.

LCKurtz said:
The Cesaro sum of a sequence gives you a new sequence which gives cumulative average of the given sequence. One place they are used is in the study of divergent sequences. For example, the sequence 1, -1, 1, -1, 1,... diverges. But, informally, you might say its "average value" is 0. And that is exactly what the Cesaro sum sequence converges to.

I understand this. It's just the technical definition that got me. Thanks!

Yes. It's 100 terms including the 1. I mistyped 101 earlier I see. I think you've got it now.

1. What is Cesaro Sum?

Cesaro Sum is a mathematical sequence that is used to determine the convergence or divergence of a series. It is named after the Italian mathematician Ernesto Cesaro.

2. How is Cesaro Sum calculated?

The Cesaro Sum is calculated by taking the arithmetic mean of the partial sums of a series. This means adding up the terms in a series and dividing by the number of terms.

3. What is the significance of understanding Cesaro Sum?

Understanding Cesaro Sum is important in determining whether a series converges or diverges. It can also be used to evaluate the sum of some infinite series.

4. How does Cesaro Sum differ from other methods of evaluating series?

Cesaro Sum differs from other methods, such as the Riemann Sum, in that it takes into account the entire sequence rather than just the limit of the partial sums. This can sometimes give a more accurate result.

5. Can Cesaro Sum be used for all types of series?

No, Cesaro Sum is only applicable to certain types of series. It is most commonly used for series that do not have a well-defined limit, such as alternating series.

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