Exploring Limits of Sequences

In summary, a sequence is a list of numbers that follow a specific pattern or rule, with each term being obtained by applying the same rule to the previous term. To determine the limit of a sequence, one can use the formal definition of a limit or various techniques such as the squeeze theorem and the ratio test. A convergent sequence is one in which the terms approach a specific value, while a divergent sequence is one in which the terms do not. In real life, exploring limits of sequences can be useful in fields such as physics, engineering, and economics for studying systems over time, analyzing growth and behavior, and evaluating stability and performance.
  • #1
kathrynag
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0

Homework Statement



I need to find if sequences have limits
1. 1,i,-1,-i,1,i,-i,1...
2. 1,i/2,-1/3,-i/4,1/5...
3. (1+i)/2,...,[(1+i)/2]^n
4.3+4i/5,...[(3+4i)/5]^n

Homework Equations





The Attempt at a Solution


I say 1 and 2 don't have limits because of the sign changes.
I say 3 and 4 have limits, but am not sure on how to find them.
3.1/2+1/2i
4.?
 
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  • #2
What tests do you have to determine if a sequence converges or not? e.g. see if it's Cauchy. Write down some, and see if they apply
 

1. What is the definition of a sequence?

A sequence is a list of numbers, called terms, that follow a specific pattern or rule. Each term in a sequence is obtained by applying the same rule to the previous term.

2. How do you determine the limit of a sequence?

To determine the limit of a sequence, you can use the formal definition of a limit, which states that a limit L is obtained when the terms of the sequence get closer and closer to L as the index of the terms gets larger and larger. You can also use various techniques such as the squeeze theorem and the ratio test to determine the limit of a sequence.

3. What is a convergent sequence?

A convergent sequence is a sequence in which the terms approach a specific value as the index of the terms increases. In other words, the limit of a convergent sequence exists.

4. What is a divergent sequence?

A divergent sequence is a sequence in which the terms do not approach a specific value as the index of the terms increases. In other words, the limit of a divergent sequence does not exist.

5. How can exploring limits of sequences be applied in real life?

Exploring limits of sequences can be applied in various fields such as physics, engineering, and economics. For example, in physics, the concept of limits of sequences is used to study the behavior of systems over time. In economics, it can be used to analyze the growth of populations or the behavior of stock prices. In engineering, limits of sequences can be used to analyze the stability and performance of systems.

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