Summing Infinite Series: Is it Acceptable?

[quasar987]

Is it generally acceptable to write the following:

a_0 + \sum_{n=1}^{\infty}a_{-n} + \sum_{n=1}^{\infty}a_{n}

as

\sum_{n=-\infty}^{\infty}a_{n}

?

 

[T@P]

i would accept it. but i speak for myself

 

[HallsofIvy]

Seems reasonable to me!

(I remember a classmate saying "by abuse of notation ... " and the professor say "lets not be that abusive!")

 

[whozum]

Series can have negative values for n? I thought the 'n' just denoted the active term?

 

[quasar987]

Haha, good quote.

 

[robert Ihnot]

I had a calculus book with that sort of stuff in it. Such a series is called a Laurent series. For example: http://mathworld.wolfram.com/LaurentSeries.html

Note last sum: \sum_{n=-\infty}^{\infty}a_{n}(z-z_{0})^n

 

[whozum]

\sum_{n=1}^{\infty}a_{-n}

I'm talking about that.. if you take a term from that sum, it is term 'a sub -1' for example, that doesn't really make sense.. you can't count -1 things..

 

[robert Ihnot]

whozum: you can't count -1 things..

If you see it that way, then you won't be writing it! But, no one said anything about counting -1 things. The subscript simply identifies the term. One might have used letters, say Greek letters like A sub Alpha.

Again, if you will look through the reference I gave, you will see a sum that runs from -infinity to -1. Are you going to claim that is not possible?
If so, argue with Wolfram Research!

 

[Lonewolf]

\sum_{n=1}^{\infty}a_{-n}

I'm talking about that.. if you take a term from that sum, it is term 'a sub -1' for example, that doesn't really make sense.. you can't count -1 things..

Well, there is a one-to-one correspondence between positive and negative integers, so you can if you're willing to be flexible with your definition of counting. There's also a one-to-one correspondence between the positive integers and integers as a whole. So, for that reason, the notation makes sense.

 

[master_coda]

Well, there is a one-to-one correspondence between positive and negative integers, so you can if you're willing to be flexible with your definition of counting. There's also a one-to-one correspondence between the positive integers and integers as a whole. So, for that reason, the notation makes sense.

Of course, the notation makes even more sense when you realize that the subscript has absolutely nothing to do with counting...

 

[whozum]

The subscript in the series term a_n denotes the term number, The first term is a_1.. the second term a_2.. the nth term a_n.. It doesn't make sense to say "the negative first term in the sequence is a_{-1}"..

To me atleast.

 

[dextercioby]

But what do you if u want to write

\sum_{n=0}^{+\infty} a_{n}\frac{1}{n^{2}+7}

and u want to include the negative values too...?

What's wring with

\sum_{-\infty}^{+\infty} a_{n}\frac{1}{n^{2}+7} ?

Daniel.

 

[Data]

I usually use the notation

\sum_{n \in \mathbb{Z}} a_n

for this. Nothing wrong with writing out the indices explicitly though.

 

[master_coda]

The subscript in the series term a_n denotes the term number, The first term is a_1.. the second term a_2.. the nth term a_n.. It doesn't make sense to say "the negative first term in the sequence is a_{-1}"..

To me atleast.

What makes you think that the a_n's are supposed to form a sequence? In fact, even if they did form a sequence, there's no reason to assume that the subscripts have to equal the term number; that's just usually convenient.

 

[whozum]

Well my experience with series and sequences terminates at Calc 2.. so I haven't dealt with them in a year.. but I was introduced with the explicit relationship that
"The first term is a_1.. the second term a_2.. the nth term a_n."

And in plain english having a 'negative xth something' doesn't make sense.. well essentially its a sum of a bunch of terms.. and with the abovep aragraph.. i guess I am oging in circles.

 

[master_coda]

Well my experience with series and sequences terminates at Calc 2.. so I haven't dealt with them in a year.. but I was introduced with the explicit relationship that
"The first term is a_1.. the second term a_2.. the nth term a_n."

And in plain english having a 'negative xth something' doesn't make sense.. well essentially its a sum of a bunch of terms.. and with the abovep aragraph.. i guess I am oging in circles.

I can see where you're saying...but that's not what subscripts mean. You can define a sequence a_1,a_2,a_3,\dotsc where the first term is a_1, then second is a_2, and so on, but it's not necessary to define a sequence that way. I can define a sequence a_1,b_1,a_2,b_2,\dotsc if that's more convenient. Or I can use any other labelling scheme I want. After all, they're just labels that I'm attaching to terms of the sequence.

So there's nothing wrong with having a sequence a_{-1},a_{-2},\dotsc. Of course, the nth term won't be a_n, but that might not be important. Or it might be important; for example, if I have the sequence a_1,b_1,a_2,b_2,\dotsc then I might want to relabel the terms as c_1,c_2,c_3,c_4,\dotsc. That doesn't change the sequence in any way, it just let's me use a more convenient notation.

And just because you're using subscripts, it doesn't automatically mean you're referring to terms of a sequence anyway. For example, I can define a function as f_c(x)=x+c, where c is any real number. Then f_1 and f_2 and f_{-0.32} and f_\pi are all functions; it's not important that it doesn't make sense for there to be a pi'th term in a sequence, because f_\pi isn't supposed to be the pi'th term of a sequence. It's just a convenient label that I chose.

 

[Hurkyl]

The general notion is of a sequence indexed by an index set I.

In the usual case with which all of you are familiar, the index set is the natural numbers, N.

The indices of your sequence are the elements of the index set. So, in a normal sequence, the indices are 0, 1, 2, ... (or 1, 2, 3, ..., depending on how you define N)

The fact that we like to write sequences as an ordered list is not part of the concept of sequence. It comes from the fact we like to have an ordering on the natural numbers. We consider a₀ is the first element of the sequence {a} precisely because we consider 0 to be the first element of the index set.

We can always choose other orderings. For instance, I may choose to order N as:

1 < 0 < 3 < 2 < 5 < 4 < ...

Then, using this ordering, the same sequence {a} would then be written:
a₁, a₀, a₃, a₂, ...


When using sequences in calculus, the chosen ordering on the index set is often important. I won't bore you with the details, but the fact we order the integers as:

... < -2 < -1 < 0 < 1 < 2 < ...

is important to the meaning of the statement

<br /> \sum_{n \ -\infty}^{\infty} a_n = \sum _{n \in \mathbb{Z}} a_n<br />

because of where the "..."s occur.

If we chose to order the integers differently, say:

... < -3 < -1 < 1 < 3 < ... < ... < -4 < -2 < 0 < 2 < 4 < ...

Then the infinite sum over the integers would acquire a different meaning.

(In particular, if the former is defined, the latter is the same, but the latter can be defined for more series)


Now, so far I've only used countable sets -- that is mildly misleading. While sequences are most commonly used with countable index sets, that is not always the case. For example, sometimes it is useful to consider a sequence whose indices range over R, or even more complicated sets! Any set whatsoever is permitted to serve as the indices.

I've also been misleading in a different way -- we don't always care about an ordering of the index set. As I mentioned, ordering is not part of the sequence concept, and there are, indeed, applications where we never bother to order the index set.

 

[master_coda]

The fact that we like to write sequences as an ordered list is not part of the concept of sequence. It comes from the fact we like to have an ordering on the natural numbers. We consider a₀ is the first element of the sequence {a} precisely because we consider 0 to be the first element of the index set.

Actually, I've almost never seen anyone call something a sequence unless it was indexed on the natural numbers. And in fact the ordering is often considered to be relevant to the concept of a sequence; even the rare generalized definitions that I've seen used have required that the index set be well-ordered.

However, I think we're basically making the same argument; most of the time, a_n is just part of an indexed set, and our indicies don't have to be from the natural numbers. Whether or not we chose to call the indexed set a sequence isn't really important.

 

[whozum]

Actually, I've almost never seen anyone call something a sequence unless it was indexed on the natural numbers. And in fact the ordering is often considered to be relevant to the concept of a sequence; even the rare generalized definitions that I've seen used have required that the index set be well-ordered.

This is probably why I'm having trouble accepting this.

Hurkyl, I feel bad having made you type all that out ot prove such a trivial point.

 

[master_coda]

Actually, I've almost never seen anyone call something a sequence unless it was indexed on the natural numbers. And in fact the ordering is often considered to be relevant to the concept of a sequence; even the rare generalized definitions that I've seen used have required that the index set be well-ordered.

This is probably why I'm having trouble accepting this.

I don't see what your point of view has to do with the remark I made. Even by the restricted definition I was using, (a_{-n})_{n\in\mathbb{N}} is still a sequence. You were arguing that it was not.

 

[whozum]

I'm not arguing that its not a sequence. I said I'm having trouble accepting that ti is because it just seems illogical to me that it would be, but hurkyl already took care ofthat.