Establish a bound on the partial sums

Oct 13, 2025 by leolavanderia.com.br 35 views

Convergence of a Product Series: A Proof of Absolute Convergence

In the realm of real analysis, sequences and series are fundamental concepts that have far-reaching implications in various branches of mathematics. One of the most crucial properties of a series is its convergence, which determines whether the series approaches a finite limit as the number of terms increases without bound. In this article, we will delve into the proof of convergence for a product series, specifically when the sum of the terms of one sequence is bounded, and the other sequence is monotone decreasing and converges to zero.

Background and Notation

Let's begin by establishing the necessary notation and background. We are given two sequences, $\{a_n\}$ and $\{b_n\}$ , and we are interested in the convergence of the product series $\sum_{i=1}^n a_nb_n$ . We are also given that $\sum_{i=1}^n b_n$ is bounded, meaning that there exists a positive real number $M$ such that $|b_n| \leq M$ for all $n$ . Additionally, we know that $a_n$ is monotone decreasing and converges to zero, i.e., $a_n \geq a_{n+1}$ for all $n$ , and $\lim_{n\to\infty} a_n = 0$ .

The Proof

To prove the convergence of the product series $\sum_{i=1}^n a_nb_n$ , we will employ the following strategy:

Establish a bound on the partial sums: We will show that the partial sums of the product series are bounded above by a convergent series.
Use the monotonicity of the sequences: We will utilize the monotonicity of the sequences $a_n$ and $b_n$ to establish a relationship between the partial sums of the product series and the partial sums of the individual sequences.
Apply the Cauchy criterion: We will use the Cauchy criterion to show that the product series converges.

Step 1: Establish a bound on the partial sums

Let $S_n = \sum_{i=1}^n a_nb_n$ be the $n$ th partial sum of the product series. We will show that $S_n$ is bounded above by a convergent series. Since $\sum_{i=1}^n b_n$ is bounded, there exists a positive real number $M$ such that $|b_n| \leq M$ for all $n$ . We can then write:

|S_n| = \left|\sum_{i=1}^n a_nb_n\right| \leq \sum_{i=1}^n |a_nb_n| \leq M \sum_{i=1}^n |a_n|

Since $a_n$ is monotone decreasing and converges to zero, the series $\sum_{i=1}^n |a_n|$ is convergent. Therefore, the partial sums $S_n$ are bounded above by a convergent series.

Step 2: Use the monotonicity of the sequences

We will now utilize the monotonicity of the sequences $a_n$ and $b_n$ to establish a relationship between the partial sums of the product series and the partial sums of the individual sequences. Let $T_n = \sum_{i=1}^n a_n$ be the $n$ th partial sum of the sequence $a_n$ . Since $a_n$ is monotone decreasing, we have:

a_n \geq a_{n+1} \geq \cdots \geq a_n

We can then write:

S_n = \sum_{i=1}^n a_nb_n \leq a_1 \sum_{i=1}^n b_n \leq a_1 M \sum_{i=1}^n 1

where $M$ is the bound on the sequence $b_n$ . Since the sequence $a_n$ converges to zero, we have:

\lim_{n\to\infty} a_n = 0

Therefore, the partial sums $S_n$ are bounded above by a convergent series.

Step 3: Apply the Cauchy criterion

We will now use the Cauchy criterion to show that the product series converges. Let $\epsilon > 0$ be given. Since the partial sums $S_n$ are bounded above by a convergent series, there exists a positive integer $N$ such that:

|S_n - S_m| < \epsilon

for all $n, m \geq N$ . This shows that the product series satisfies the Cauchy criterion, and therefore, it converges.

Conclusion

In this article, we have proven that the product series $\sum_{i=1}^n a_nb_n$ converges when the sum of the terms of one sequence is bounded, and the other sequence is monotone decreasing and converges to zero. We have employed the following strategy:

Establish a bound on the partial sums: We have shown that the partial sums of the product series are bounded above by a convergent series.
Use the monotonicity of the sequences: We have utilized the monotonicity of the sequences $a_n$ and $b_n$ to establish a relationship between the partial sums of the product series and the partial sums of the individual sequences.
Apply the Cauchy criterion: We have used the Cauchy criterion to show that the product series converges.

In our previous article, we proved that the product series $\sum_{i=1}^n a_nb_n$ converges when the sum of the terms of one sequence is bounded, and the other sequence is monotone decreasing and converges to zero. In this article, we will address some common questions and concerns related to this result.

Q: What are the conditions for the convergence of the product series?

A: The product series $\sum_{i=1}^n a_nb_n$ converges when the sum of the terms of one sequence is bounded, and the other sequence is monotone decreasing and converges to zero.

Q: What does it mean for a sequence to be monotone decreasing?

A: A sequence $\{a_n\}$ is said to be monotone decreasing if $a_n \geq a_{n+1}$ for all $n$ . This means that the terms of the sequence are non-increasing.

Q: What does it mean for a sequence to converge to zero?

A: A sequence $\{a_n\}$ converges to zero if $\lim_{n\to\infty} a_n = 0$ . This means that the terms of the sequence approach zero as $n$ approaches infinity.

Q: How do we establish a bound on the partial sums of the product series?

A: We can establish a bound on the partial sums of the product series by using the fact that the sum of the terms of one sequence is bounded. Specifically, we can write:

|S_n| = \left|\sum_{i=1}^n a_nb_n\right| \leq \sum_{i=1}^n |a_nb_n| \leq M \sum_{i=1}^n |a_n|

where $M$ is the bound on the sum of the terms of one sequence.

Q: How do we use the monotonicity of the sequences to establish a relationship between the partial sums of the product series and the partial sums of the individual sequences?

A: We can use the monotonicity of the sequences to establish a relationship between the partial sums of the product series and the partial sums of the individual sequences by writing:

S_n = \sum_{i=1}^n a_nb_n \leq a_1 \sum_{i=1}^n b_n \leq a_1 M \sum_{i=1}^n 1

where $M$ is the bound on the sum of the terms of one sequence.

Q: How do we apply the Cauchy criterion to show that the product series converges?

A: We can apply the Cauchy criterion by showing that the partial sums of the product series satisfy the Cauchy criterion. Specifically, we can write:

|S_n - S_m| < \epsilon

for all $n, m \geq N$ , where $\epsilon > 0$ is given and $N$ is a positive integer.

Q: What are some common applications of the convergence of a product series?

A: The convergence of a product series has far-reaching implications in various branches of mathematics, including real analysis, sequences and series, and functional analysis. Some common applications include:

Convergence of power series: The convergence of a power series can be established using the convergence of a product series.
Convergence of Fourier series: The convergence of a Fourier series can be established using the convergence of a product series.
Convergence of integral transforms: The convergence of an integral transform can be established using the convergence of a product series.

Conclusion

In this article, we have addressed some common questions and concerns related to the convergence of a product series. We have established the conditions for the convergence of the product series, and we have provided a step-by-step guide to establishing a bound on the partial sums of the product series, using the monotonicity of the sequences, and applying the Cauchy criterion. We have also discussed some common applications of the convergence of a product series.