Understanding about Sequences and Series

In summary, there are various methods to determine if a sequence or series converges or diverges, including the use of geometric and telescoping series and comparison with known convergent or divergent series. Complex analysis can also be a helpful tool in studying Taylor series. Ultimately, determining convergence depends on the specific properties of the series in question.
  • #1
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Homework Statement:: Tell me if a sequence or series diverges or converges
Relevant Equations:: Geometric series, Telescoping series, Sequences.

If I have a sequence equation can I tell if it converges or diverges by taking its limit or plugging in numbers to see what it goes too?

Also if I have a series, can I tell if it converges or diverges if it goes to a certain number? Or does it depend on the type of series? I know the geometric series method is a/1-r and the telescoping series is the first value of the first term subtracted by the last value in the last term. Then you take the limit to see what it goes to. I want to understand how you can tell if it nonmonotonic and bounded and also if it converges or diverges.

Im confused on methods to use and so far I have only learned Geometric series, telescoping series and harmonic series in class.

[Moderator's note: moved from a technical forum.]
 
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  • #2
The definition says: If the sequence of partial sums ##S_n=\sum_{k=1}^n a_k## converges, then the series converges. That means ##\lim_{n \to \infty} S_n < \infty \Longrightarrow \sum_{k=1}^\infty a_k < \infty \,.## This may look trivial, but it transforms the convergence of a sum into a convergence of a sequence that is easier to determine. If you plug in some numbers, then you have to do it for the sequence. The standard example is the series ##\sum_{k=1}^\infty 1/n## which diverges because ##\ln n =\int_1^n (1/x) dx## goes to infinity and our series is basically the Riemann sum of the integral. So, plugging in numbers won't give you enough information. It can be used to get a feeling, a heuristic, but even this could set you on the wrong track.

There are a couple of criteria to determine whether a series converges or not:
https://en.wikipedia.org/wiki/Convergence_tests
or the nice list in table form here:
https://de.wikipedia.org/wiki/Konvergenzkriterium#Konvergenzkriterien_für_Reihen
 
  • #3
There are a variety of techniques to test if a series converges. A lot of it is like a bag of tricks, comparing the series in question with a series that is known to converge or not.
A lot of number series can be compared to a Taylor series ##\sum{a_n x^n}## with a specific value of ##x## for which the convergence properties are known. Complex analysis is a very helpful and methodical way to study the Taylor series.
 
  • #4
"Does this converge?" is generally much easier to answer than the follow up "if so, what does it converge to?"
 
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Related to Understanding about Sequences and Series

1. What is a sequence?

A sequence is a list of numbers or objects that follow a specific pattern or rule. Each element in the sequence is called a term, and the position of a term in the sequence is called its index.

2. What is the difference between a sequence and a series?

A sequence is a list of terms, while a series is the sum of those terms. In other words, a series is the result of adding up all the terms in a sequence.

3. What is the difference between an arithmetic and geometric sequence?

An arithmetic sequence is a sequence in which each term is obtained by adding a constant value to the previous term. A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a constant value.

4. How do you find the nth term of a sequence?

To find the nth term of a sequence, you need to identify the pattern or rule that the sequence follows. Once you have identified the pattern, you can use it to find the value of any term in the sequence by plugging in the term's index into the formula.

5. How can sequences and series be applied in real life?

Sequences and series have many real-life applications, such as in financial investments, population growth, and natural phenomena. They can also be used in computer algorithms, music composition, and sports statistics.

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