Question about index of notation

[mesa]

I see summation examples in my textbooks often give something where there is an a_i 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 a_m+a_m+1+...+a_n-1+a_n
What mathematical representation are they trying to convey with the letters m and n under and to the right of a?

 

[SteamKing]

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.

 

[mesa]

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.

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

 

[jbunniii]

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

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.

 

[mesa]

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.

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 a_i?
Are there other places where functions are just written as 'a' or is it exclusive to summations?

 

[jbunniii]

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 a_i?

Because $a$ is the function. What does $f$ even refer to here?

Are there other places where functions are just written as 'a' or is it exclusive to summations?

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, $a(1) + a(2) + a(3) + a(4) + a(5)$.

 

[mesa]

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, $a(1) + a(2) + a(3) + a(4) + a(5)$.

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?

 

[jbunniii]

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?

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.

 

[mesa]

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.

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

 

[SteamKing]

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

 

[jbunniii]

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

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 $\mathbb{R}$.

 

[mesa]

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

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 ;)

 

[jbunniii]

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 ;)

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.

 

[mesa]

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.

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 :)