Series and Sequences

[DisaFear]

Hi guys,

There is one part of my mathematics unit that I do not understand at all.

That is infinite series.

I understand how sequences work. It's like a counting game, with a pattern. 1, 2, 3, 4, 5, 6, 7... (right?)

Now, I understand the meaning of a series (I think!). We are just adding each term in some sequence we randomly obtained, right? Is that what a series is? What is a 'sequence of partial sums' and how does this relate to series? How does one obtain a 'sequence of partial sums'?
If I recall all the reading I've been doing, and what a few peps have told me, the 'series' is the LIMIT of the 'sequence of partial sums'. Is that true?

But I need to understand what this 'sequence of partial sums' is first and where we get this!!

I understand that sequences can be convergent or divergent. Yea...thought I'd mention that Is this the thing that defines a series?

What does the geometric series signify? What does the power series signify?

Do we obtain an 'infinite series' from the 'sequence of partial sums' or the other way around?

Thanks for any help for this seemingly easy topic, much appreciated.

My poor head 

[pi]

I'll go from GMA knowledge for now, and then TT or someone can pick up from where uni comes into it

Essentially you are correct in definitions, a sequence is just a list ordered numbers and a series is the sum of the terms in any given sequence.

A partial sum is the sum of a certain number of terms of any given sequence, and a sequence of these partial sums is what the name suggests, a sequence with each term being a partial sum, so for example . As for "'series' is the LIMIT of the 'sequence of partial sums'", yes that is correct because the sequence of the partial sums is essentially a new sequence all together. And the "series", as defined by being the sum of all the terms in any given sequence (a somewhat rudimentary definition, but let's work with it), will end up being the term , which correlates to your statement of it being the limit of (which as aforementioned is the sequence of partial sums of the original sequence).

As far as I know, convergence is one of many properties a series can have. But someone else is better suited to explain the other properties in depth here.

A geometric series, by definition, is the sum of a geometric sequence (or progression): one where there is a common ration between terms.

I'm not cut out to explain the rest of your post with my knowledge haha, but I hope this covered some of the basics

[DisaFear]

It is starting to make much more sense, thanks.

So is a 'sequence of partial sums' a 'sequence within a sequence' or a 'sequence of a sequence'?

Could anyone put up a simple geometric series, and show how one goes about finding if it is convergent/divergent?
I'm guessing this is what the integral test/root test/comparision test/ratio test/"enter lots of tests here" are for.

A power series explanation/example would be great too.

Thanks all

[kamil9876]

So is a 'sequence of partial sums' a 'sequence within a sequence' or a 'sequence of a sequence'?

Neither: If you are given some sequence of numbers then we can define the nth partial sum to be i.e the sum of the first terms. The we can consider a new sequence of numbers.

Now what you are always interested in is whether has a limit i.e if the partial sums have a limit. If so then we say that the series converges.

Could anyone put up a simple geometric series, and show how one goes about finding if it is convergent/divergent?

For geometric series there is a simple way of determining whether it converges or not. converges when and it does not converge otherwise. Let's prove this straight from the definition I gave above:

So we first have to find the nth partial sum.



Where I used a well known formula you may be familiar with from sometime in your education.

Now let's check whether the the sequence does converge. Let's firstly consider the case . You know that if hence . Hence this is the limit of the 's and so by definition the series converges to the value .

If on the other hand in the case can you see why the does not converge to any number?

I'm guessing this is what the integral test/root test/comparision test/ratio test/"enter lots of tests here" are for.

As my post is long enough as it is, I won't go into more details, what I've written so far seems to be enough to think about for the moment. You can check TT's Maths Thread, I remember he went through heaps of these ages ago ( forum search ratio test, root test etc. since it is a large thread so it may be hard to find where exactly)

[DisaFear]

Neither: If you are given some sequence of numbers then we can define the nth partial sum to be i.e the sum of the first terms. The we can consider a new sequence of numbers.

Now what you are always interested in is whether has a limit i.e if the partial sums have a limit. If so then we say that the series converges.

Big thanks, almighty kamil.

Alright. So.
We have a sequence of terms.
By adding 'n' terms of that sequence together, we get the partial sum sequence.

By S₃, you mean adding the first three terms right? And so on for the others

By "S₁, S₂, S₃...", you sort of mean S_n do you? (As it goes towards'ish thingy)

We have S₁, S₂, S₃. What we are interested in is this one: S_n (right?)

(It was originally the S₁, S₂, S₃ that got me confused in my readings. I thought we were adding THESE together to get the series - which raised the question of repeated terms and why do that)

Thanks so much!

[pi]

By S₃, you mean adding the first three terms right? And so on for the others

Yep

By "S₁, S₂, S₃...", you sort of mean S_n do you? (As it goes towards'ish thingy)

We have S₁, S₂, S₃. What we are interested in is this one: S_n (right?)

Yep, S_n is technically the "series" (not a partial sum)