Homework Help: Question about index of notation

1. Feb 18, 2013

mesa

I see summation examples in my text books often give something where there is an ai next to the summation itself. It also shows the index of notation to be i=m at the bottom an m at the top with the whole thing equal to am+am+1+...+an-1+an
What mathematical representation are they trying to convey with the letters m and n under and to the right of a?

2. Feb 18, 2013

SteamKing

Staff Emeritus
The subscripts (m, n, etc.) are there to identify a particular a-value in the expanded summation sequence.

In your example, a-sub m is the mth a value, the next is the m+1 th value, ..., all the way to the n-th value. The numeric values of m and n are determined separately.

3. Feb 18, 2013

mesa

It's this 'a' value that is tripping me up. What does am mean in mathematical notation? Or are you saying 'a' just a placeholder and the n, and m are the functions?

4. Feb 18, 2013

jbunniii

It is the opposite. $a$ is the function, and $m$ and $n$ are arguments. In the usual function notation we would write $a(m)$, $a(n)$, etc. When the function takes only integer arguments, it is a convention to write $a_m$ instead of $a(m)$, but they mean the same thing.

5. Feb 18, 2013

mesa

Okay, so m and n are just representations for each piece of the summation of the function 'a'. Never had a problem doing the actual summations but this notation has been driving me nuts each time I look at it when referencing my texts.

So why isn't it written as f(a)i instead of ai?
Are there other places where functions are just written as 'a' or is it exclusive to summations?

6. Feb 18, 2013

jbunniii

Because $a$ is the function. What does $f$ even refer to here?
We can use any letter we like for functions, including both the roman and greek alphabets. All the notation $a_n$ means is that we have some function $a$, which can be evaluated for integer values $n$, and instead of writing $a(n)$ we write $a_n$.

If we have a summation
$$\sum_{n = 1}^{5} a_n$$
that simply means $a_1 + a_2 + a_3 + a_4 + a_5$, or equivalently, $a(1) + a(2) + a(3) + a(4) + a(5)$.

7. Feb 18, 2013

mesa

It's just odd compared to what I have seen so far. Is representing functions with just a letter a standard for mathematics or unique to just summations?

8. Feb 18, 2013

jbunniii

How do you usually represent a function? Wouldn't $f$ be a typical example, where the value of the function evaluated at $x$ is $f(x)$? So all we are doing is saying that if $x$ is restricted to integer values, it's a standard convention (but by no means required) to write it as $f_x$ instead of $f(x)$. It is also a convention to use a letter like $n$ instead of $x$ to indicate that it only takes on integer values. But this is also not a requirement.

In your case, we are using the letter $a$ instead of the letter $f$, and we are using $n$ for the argument instead of $x$. But the notation all means the same thing.

9. Feb 18, 2013

mesa

Very good with the explanation; you 'summed' it up nicely ;)

10. Feb 18, 2013

SteamKing

Staff Emeritus
The a values don't necessarily represent functions. The series of a values could also be different constants, as well.

11. Feb 18, 2013

jbunniii

But $a$ itself is a function. For example, if $a_1, a_2, a_3, \ldots$ is a real-valued sequence, then $a : \mathbb{N} \rightarrow \mathbb{R}$ is a function.

The values of $a$ can be functions, of course. For example, we may have $a_n(x) = \sin(nx)$. Then $a : \mathbb{N} \rightarrow X$, where $X$ is the set of real-valued functions with domain $\mathbb{R}$.

12. Feb 18, 2013

mesa

The constants are just pulled out of the summation correct? But technically it can serve as a representation of those as well; basically anything you choose to find the sum of for n# of integers up to m based on our function 'a', right?

It's interesting going back through the books, I haven't looked at these in awhile and the representation in the texts using '$a$$i$' (neat way of writing these by the way jbunniii, I'll be using $in the future) always gave me trouble for some reason... must be my age, ha ha ha. Be grateful you were wise enough to tackle these subjects when you were still young ;) 13. Feb 18, 2013 jbunniii It's just a question of getting used to the notation. A quick anecdote: group theorists, especially old-fashioned ones, like to write their functions to the right of the argument, i.e., they write$(x)f$when the rest of the world would write$f(x)$. (They have their reasons for doing this, which I won't go into.) I recall reading one author defending this practice, by noting that we use all sorts of idiosyncratic functional notation without even thinking of it. For example,$e^x$(function at the base, argument in the exponent),$x^2$(argument at the base, function in the exponent),$n!## (argument to the left), and so forth. He certainly had a valid point.

14. Feb 18, 2013

mesa

It seems like we get accommodated with these notations with time, I just need to 'get used to' the rest of it; all that is left is everything :)