Understanding about Sequences and Series

[Vividly]

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.]

 

[fresh_42]

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

 

[FactChecker]

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.

 

[pasmith]

"Does this converge?" is generally much easier to answer than the follow up "if so, what does it converge to?"