# Sequences and their limits, convergence, supremum etc.

• MHB
• ertagon2
In summary, a sequence is a list of numbers that follow a specific pattern or rule and are written in a specific order. The limit of a sequence is the value that the terms approach as the number of terms increases towards infinity. To determine if a sequence converges or diverges, you need to see if the terms approach a finite value or not. The supremum of a sequence is the smallest upper bound of the sequence. Sequences are commonly used in science to model and analyze real-world phenomena, such as population growth and musical vibrations.
ertagon2
Could someone check if my answers are right and help me with question 5?
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Are you sure about the third statement in 4?
Also, you probably have some idea about 5. Are you doubting between two possibilities?

Last edited:

Sure, I'd be happy to help! Can you provide more information about the question and your answers for question 5? That way, I can check your work and provide any necessary assistance. Looking forward to hearing back from you!

## 1. What is a sequence?

A sequence is a list of numbers that follow a specific pattern or rule. The numbers in a sequence are usually denoted by a subscript and are written in a specific order.

## 2. What is the limit of a sequence?

The limit of a sequence is the value that the terms in the sequence approach as the number of terms increases towards infinity. This means that as the terms in the sequence get closer and closer to the limit, the difference between them becomes smaller and smaller.

## 3. How do you determine if a sequence converges or diverges?

A sequence converges if its terms approach a finite value (the limit) as the number of terms increases towards infinity. On the other hand, a sequence diverges if its terms do not approach a finite value and instead either increase or decrease without bound.

## 4. What is the supremum of a sequence?

The supremum of a sequence is the smallest number that is greater than or equal to all of the numbers in the sequence. In other words, it is the upper bound of the sequence.

## 5. How can sequences be used in real life?

Sequences are used in various areas of science, such as physics, engineering, and computer science, to model and analyze real-world phenomena. For example, the Fibonacci sequence is used to describe the growth of populations in biology, while the harmonic sequence is used to analyze the vibrations of musical instruments in physics.

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