# Cesaro summability implies bounded partial sums

• stripes
In summary: Since we are given that \sum_{n=0}^{\infty} c_n is Cesaro summable, we know that this limit exists and is equal to the Cesaro sum of the series. Therefore, the partial sums are bounded by this limit, and thus are bounded. In summary, if \sum ^{\infty}_{n=0} c_{n} is Cesaro summable, then the partial sums S_{N} are bounded by the limit \lim_{n \to \infty} S_n.
stripes

## Homework Statement

Suppose $c_{n} > 0$ for each $n\geq 0.$ Prove that if $\sum ^{\infty}_{n=0} c_{n}$ is Cesaro summable, then the partial sums $S_{N}$ are bounded.

--

## The Attempt at a Solution

I tried contraposition; that was getting me nowhere. I have a few inequalities here and there but they don't tell me anything. I need to show that there exists an upperbound for the partial sums. This means there exists a least upperbound. I need to find that least upperbound. Because$c_{n} > 0$ for each $n\geq 0,$ then the series is nondecreasing, which means the partial sums are nondecreasing, so we are looking for an upperbound, not a lowerbound.

stripes said:

## Homework Statement

Suppose $c_{n} > 0$ for each $n\geq 0.$ Prove that if $\sum ^{\infty}_{n=0} c_{n}$ is Cesaro summable, then the partial sums $S_{N}$ are bounded.

--

## The Attempt at a Solution

I tried contraposition; that was getting me nowhere. I have a few inequalities here and there but they don't tell me anything. I need to show that there exists an upperbound for the partial sums. This means there exists a least upperbound. I need to find that least upperbound.

That least upper bound, if it exists, is $\lim_{n \to \infty} S_n$, which is the definition of $\sum_{n=0}^{\infty} c_n$ in the traditional sense.

## 1. What is Cesaro summability?

Cesaro summability is a mathematical concept that is used to determine the convergence of a series. It takes into account the average of the partial sums of the series, rather than just the individual terms. In simple terms, it is a way to determine whether a series is convergent or not.

## 2. How does Cesaro summability imply bounded partial sums?

If a series is Cesaro summable, it means that the average of its partial sums is finite. This implies that the series must have bounded partial sums, as the average of the partial sums cannot be infinite if the individual terms are finite. This is a useful property, as it allows us to make conclusions about the convergence of a series based on its Cesaro summability.

## 3. What is the significance of bounded partial sums in Cesaro summability?

The boundedness of partial sums in Cesaro summability allows us to make stronger statements about the convergence of a series. If the partial sums are bounded, it means that the series is convergent, and we can determine its limit using the Cesaro summation method. This makes Cesaro summability a powerful tool in analyzing series.

## 4. Can Cesaro summability be applied to all series?

No, Cesaro summability is not applicable to all series. It only works for series that satisfy certain conditions, such as having finite individual terms and a well-defined limit. Furthermore, it may not always give the same result as other methods of determining convergence, so it is important to use it in conjunction with other techniques.

## 5. How is Cesaro summability used in real-world applications?

Cesaro summability has many applications in mathematics, engineering, and physics. It is used to analyze the convergence of infinite series in many fields, such as signal processing, control theory, and numerical analysis. It is also used to determine the stability of systems in engineering and physics, making it a valuable tool in practical applications.

Replies
1
Views
906
Replies
1
Views
5K
Replies
2
Views
2K
Replies
12
Views
2K
Replies
3
Views
1K
Replies
2
Views
4K
Replies
7
Views
1K
Replies
3
Views
2K
Replies
1
Views
1K
Replies
1
Views
1K