Thread: Series

I was just studying the series, and there were two terms, convergence and divergence, and then more like absolute convergence etc... came up.

now i know how to prove its convergent and stuff but i dont get the definition behind it.
could anyone explain it??
thank you

Originally Posted by Heinsbergrelatz

I was just studying the series, and there were two terms, convergence and divergence, and then more like absolute convergence etc... came up.

now i know how to prove its convergent and stuff but i dont get the definition behind it.
could anyone explain it??
thank you

A series converges if the sequence of partial sums converges.

It converges absolutely if the series of absolute values of the original series converges.

A series diverges if it does not converge.

How can you possibly prove that a series is convergent if you don't understand what "convergent " means ? This sounds as though you are mindlessly pushing symbols. There is no point in doing that.

I am assuming that you know what it means for a sequence to converge and what the sequence of partial sums is. If not, go back and review those terms.

There are other, fancier ways to approach series, called summability methods, but first you need to understand what basic convergence means.

when i mean prove, i actually mean being able to solve questions that ask if the series converges or not. But what i actually wanted to ask was, when i solve these questions, sometimes i get lost in the definition of what the terms actually mean.

Originally Posted by Heinsbergrelatz

when i mean prove, i actually mean being able to solve questions that ask if the series converges or not. But what i actually wanted to ask was, when i solve these questions, sometimes i get lost in the definition of what the terms actually mean.

In that case you have not clearly posed your question.

You are also giving the impression that you are simply pushing symbols according to some mystical rules to determine if a series converges without understanding what convergence means. That is a waste of time.

You are also giving the impression that you are simply pushing symbols according to some mystical rules to determine if a series converges without understanding what convergence means. That is a waste of time

i was on series, and i thought i was comfortable with it, so i decided to open up IB past papers, (the thing i will be studying in two years time), and suddenly found the term convergence. Never seen convergence in my textbook before.

Originally Posted by Heinsbergrelatz

You are also giving the impression that you are simply pushing symbols according to some mystical rules to determine if a series converges without understanding what convergence means. That is a waste of time

i was on series, and i thought i was comfortable with it, so i decided to open up IB past papers, (the thing i will be studying in two years time), and suddenly found the term convergence. Never seen convergence in my textbook before.

In that case what you need to do is learn what convergence means.

It is a very important concept.

Do not start with series. That is secondary.

Start with convergence of sequences. Once you understand that, then and only then are you ready to look at the convergence and divergence of series.

I have no idea how you could possibly be comfortable with "series" if you don't understand what convergence is.

I'm sure a lot of what he's seen is symbol manipulation in textbooks, and little actually in depth explanation.

In that case what you need to do is learn what convergence means.

It is a very important concept.

Do not start with series. That is secondary.

Start with convergence of sequences. Once you understand that, then and only then are you ready to look at the convergence and divergence of series.

I have no idea how you could possibly be comfortable with "series" if you don't understand what convergence is

strangely enough the textbook i use does not have any implications on anything that has got to do with convergence. Its just harmonic, arithmetic and geometric. The applications based on these questions did not seem too hard ill say, so i was feeling alittle comfortable with it. The only proofs included is, how one arrived to the geometric ad arithmetic progression.
But luckily enough convergence had got to do with limits and continuity, and michael spivak's book on calculus helped me alot on understanding the idea of limits and its properties, though it was very tough i would have to say. (VERY)

I'm sure a lot of what he's seen is symbol manipulation in textbooks, and little actually in depth explanation.

Yes you are very right, and one day i got so fed up with the idea that i am not quite understanding the definition of what i am doing, so i asked my teacher once, "Sir can you prove the Quotient rule?" and my teacher was blank, (he explained the very next day....) . this kind of showed me that secondary mathematics based on pure intuition with not much rigor proof or precise understanding.
thats when i decided to study mathematics alone at home, ignoring the school's coursework and curriculum.