Problem with substitution

[Foreverlearning]

Hello, I have a problem and I hope some one could help. This is not a homework, I am trying to learn some concepts.
Please check the attachment. Thanks much.

 

[Otis]

EDIT: I misconstrued the function before posting the following.

Hi. I'm not sure what you mean by saying f(a) represents two different values, but are you thinking that function f is a constant function because they wrote f(a)=0?

That equation says that f will output the value zero, when the input value is x=a. It does not say anything about the function's output values when x is not the number a.

 

[Dr.Peterson]

Hello, I have a problem and I hope some one could help. This is not a homework, I am trying to learn some concepts.
Please check the attachment. Thanks much.

Can you show us the original problem as given to you? What you show makes no sense:

​

Since x is the variable over which the summation is taken, it is not an argument of the function; the scope of x is inside the summation.

I'm guessing it may have been meant to be something like this:

​

But even that would be odd; and as written, you are right that f(x) is constant, to the extent that it is valid to use x for two different variables in the same line. The RHS is independent of x.

 

[Steven G]

I 2nd that this problem makes no sense.
The rhs is a function of n while the lhs is a function of x

 

[Otis]

f(x) is constant, to the extent that it is valid to use x for two different variables in the same line

This function is not the form that I thought it was. (I'd assumed too much.)

 

[Foreverlearning]

Hi. I'm not sure what you mean by saying f(a) represents two different values, but are you thinking that function f is a constant function because they wrote f(a)=0?

That equation says that f will output the value zero, when the input value is x=a. It does not say anything about the function's output values when x is not the number a.

You say

I'm not sure what you mean by saying f(a) represents two different values

It is clear that In (1) f(a) is zero, in (2), it has a non-zero value.

You say

"it does not say anything about the function output when x is not the number a".

The expression on the R.H.S of (1) describes the relationship precisely for any "x". The problem is with x=a. What more to say?

 

[Foreverlearning]

Can you show us the original problem as given to you? What you show makes no sense:

View attachment 38109​

Since x is the variable over which the summation is taken, it is not an argument of the function; the scope of x is inside the summation.

I'm guessing it may have been meant to be something like this:

View attachment 38108​

But even that would be odd; and as written, you are right that f(x) is constant, to the extent that it is valid to use x for two different variables in the same line. The RHS is independent of x.

The problem results from a personal encounter. It is not in a book.
The problem statement is accurate.
Using x as an index to the summation and in the object of the summation is definitely valid. Example: Taylor Expansion.

 

[Foreverlearning]

I 2nd that this problem makes no sense.
The rhs is a function of n while the lhs is a function of x

You say:

The rhs is a function of n while the lhs is a function of x

How about f(x)=n for any number n? It is not "no sense"!

 

[Otis]

What more to say?

Hi. I think we cross-posted yesterday (the timestamps differ by 24min), so you may have missed seeing my previous post. I apologize for misreading the exercise statement in your op.

 

[Steven G]

How can x be changing on the rhs from 1 to n yet on the lhs it is one value?
x is being used to represent two values which is never good.

 

[Dr.Peterson]

The problem results from a personal encounter. It is not in a book.
The problem statement is accurate.

In other words, presumably the person who gave it to you is responsible for the difficulty in it, however we describe it. What did that person think of it?

Using x as an index to the summation and in the object of the summation is definitely valid. Example: Taylor Expansion.

As I said, "to the extent that it is valid to use x for two different variables in the same line, ..."; it can be taken as valid, but, in my opinion at least, it is risky, because it is easy to misread (as @Otis, I believe, did -- and I did initially). The RHS uses x differently than the LHS, leading to the confusion I think all of us are feeling.

You presumably have in mind one particular presentation of the Taylor expansion; can you show that to us?

What I typically see is something like this:

Taylor series - Wikipedia

en.wikipedia.org

The Taylor series of a real or complex-valued function f (x), that is infinitely differentiable at a real or complex number a, is the power series​

Here, n! denotes the factorial of n. The function f(n)(a) denotes the nth derivative of f evaluated at the point a.​

Here the summation is over n, which is not used with a different meaning elsewhere in the equation. The series represents f(x), and x is clearly an input to the RHS, as it is not in your summation.

Looking for discussions of the scope of the index of a summation, I found this, which says,

​

If we do the same with your summation, we would rewrite [math]\sum_{x=1}^{n}(x-a)[/math] to [math]\sum_{i=1}^{n}(i-a)[/math] which makes it clearer that x as used on the LHS is not present.

So, as I said, if we consider your notation valid, then the RHS is in fact, a function of a and n, not of x; my statement that it is constant means that it is independent of x, and depends only on the parameters a and n. Whatever value x might have before evaluating the RHS, it is overridden during evaluation, being assigned values 1, 2, ..., n.

That makes your initial statement incorrect: the terms (a-x) are not all equal to zero, because x is different in each of them, and is not equal to whatever value you put into "f(x)".

 

[Foreverlearning]

In other words, presumably the person who gave it to you is responsible for the difficulty in it, however we describe it. What did that person think of it?

As I said, "to the extent that it is valid to use x for two different variables in the same line, ..."; it can be taken as valid, but, in my opinion at least, it is risky, because it is easy to misread (as @Otis, I believe, did -- and I did initially). The RHS uses x differently than the LHS, leading to the confusion I think all of us are feeling.

You presumably have in mind one particular presentation of the Taylor expansion; can you show that to us?

What I typically see is something like this:

Taylor series - Wikipedia

en.wikipedia.org

The Taylor series of a real or complex-valued function f (x), that is infinitely differentiable at a real or complex number a, is the power series​

Here, n! denotes the factorial of n. The function f(n)(a) denotes the nth derivative of f evaluated at the point a.​

Here the summation is over n, which is not used with a different meaning elsewhere in the equation. The series represents f(x), and x is clearly an input to the RHS, as it is not in your summation.

Looking for discussions of the scope of the index of a summation, I found this, which says,

View attachment 38115​

If we do the same with your summation, we would rewrite [math]\sum_{x=1}^{n}(x-a)[/math] to [math]\sum_{i=1}^{n}(i-a)[/math] which makes it clearer that x as used on the LHS is not present.

So, as I said, if we consider your notation valid, then the RHS is in fact, a function of a and n, not of x; my statement that it is constant means that it is independent of x, and depends only on the parameters a and n. Whatever value x might have before evaluating the RHS, it is overridden during evaluation, being assigned values 1, 2, ..., n.

That makes your initial statement incorrect: the terms (a-x) are not all equal to zero, because x is different in each of them, and is not equal to whatever value you put into "f(x)".

Thank you for the time and effort you have put in the answer. In fact I came across this expression during my self study (I am retired). My reference to Taylor's series was inaccurate. I don't want to change the index summation to a vriable other than "x" (that would take the fun away). I see that to represent the sum of (1-a), (2-x), (3-a) and (4-a), it makes sense to use this form:


In fact the above form is what is used in programming (for loops, do loops and Cobol's Perform Varying construct).

I appreciate the point you made above (with a reference too (cool)), that this confusing. You are correct it is. My entire point is to learn. I have never encountered a similar case or an issue with the function expression definition before!

Thanks again for your help.

 

[Dr.Peterson]

Thank you for the time and effort you have put in the answer. In fact I came across this expression during my self study (I am retired).

So this wasn't just made up by a friend, but was in a book or other source? What was it?

I don't want to change the index summation to a variable other than "x" (that would take the fun away).

If the point is to make it tricky, then you accomplished the goal. Usually, we try to be clear.

In fact the above form is what is used in programming (for loops, do loops and Cobol's Perform Varying construct).

Yes, some languages do allow you to use the same variable name in different scopes (such as within a loop), though others don't. It's generally not recommended when it can lead to confusion.

But the important thing is that your initial statement was wrong:

​

Do you acknowledge this? And that (very natural) error was due to the reuse of a variable name outside of its scope. That's why this is not a good thing to "learn".

 

[Foreverlearning]

In other words, presumably the person who gave it to you is responsible for the difficulty in it, however we describe it. What did that person think of it?

As I said, "to the extent that it is valid to use x for two different variables in the same line, ..."; it can be taken as valid, but, in my opinion at least, it is risky, because it is easy to misread (as @Otis, I believe, did -- and I did initially). The RHS uses x differently than the LHS, leading to the confusion I think all of us are feeling.

You presumably have in mind one particular presentation of the Taylor expansion; can you show that to us?

What I typically see is something like this:

Taylor series - Wikipedia

en.wikipedia.org

The Taylor series of a real or complex-valued function f (x), that is infinitely differentiable at a real or complex number a, is the power series​

Here, n! denotes the factorial of n. The function f(n)(a) denotes the nth derivative of f evaluated at the point a.​

Here the summation is over n, which is not used with a different meaning elsewhere in the equation. The series represents f(x), and x is clearly an input to the RHS, as it is not in your summation.

Looking for discussions of the scope of the index of a summation, I found this, which says,

View attachment 38115​

If we do the same with your summation, we would rewrite [math]\sum_{x=1}^{n}(x-a)[/math] to [math]\sum_{i=1}^{n}(i-a)[/math] which makes it clearer that x as used on the LHS is not present.

So, as I said, if we consider your notation valid, then the RHS is in fact, a function of a and n, not of x; my statement that it is constant means that it is independent of x, and depends only on the parameters a and n. Whatever value x might have before evaluating the RHS, it is overridden during evaluation, being assigned values 1, 2, ..., n.

That makes your initial statement incorrect: the terms (a-x) are not all equal to zero, because x is different in each of them, and is not equal to whatever value you put into "f(x)".

Thanks for your reply. As I mentioned, I came across this while I was studying and trying to self-study. No book, and and no friends. I am not trying to be tricky or to confuse anyone. I am merely trying to learn. I was surprised that this form was never shown to me over years of learning, and was not sure how to avoid such a mistake.