@vv 

I know, but this was not my purpose. I only wanted to provide a solution which requires no explicit loop (even if on might comsider  map and "~" are disguised loops).
In my mind it was just a kind of stylistic exercise.

@Scot Gould 

What I like about your videos is that you do the course live, filling in the worksheet step by step.

I myself have developed a series of courses for corporate training courses on statistics which are based on the use of Maple (which implies that people already have some knowledge of Maple).
And the main question is "Is it better to build the worksheet (or document) live, or to run a step-by-step worksheet that's already been written (as Robert Lopez does)?"
I feel that your choice is well suited to short videos and I wonder whether it would be appropriate for one-hour training sessions, for example.

@Scot Gould 

I didn't know at all about this site, it's an excellent initiative and I will promote it around me.

@acer 

I would have thought the opposite.
Thanks for the reply

@dharr 

Ok, the "it is the coefficient in the expansion (1+x)^alpha = sum(binomial(alpha,k)*x^k,k=0..infinity)" definition makes sense as long as do not wanr to draw any parallel to the factorials (IMO).

An opinion confirmed by this source Binomial_Theorem :
In section Generalizations > Newton's generalized binomial theorem
"In order to do this, one needs to give meaning to binomial coefficients with an arbitrary upper index, which cannot be done using the usual formula with factorials. However, for an arbitrary number r, one can define ..."

At last, the Maple: dictionnary (entry binomial coefficient) says more or less the same thing than the source above.

Thanksagain for your precious comments.

@dharr 

Thank you @dharr for this inquiry which must have taken up a lot of your time.

I have carefully reas and read again your reply and I still do not understand if 0

1. results from a decision not to return an error but something like "caution, the result "0" is a convention when b < a",
2. or a mathematical extension of the definition of binomial(a, b) for b < a?

As I mentioned earlier, I couldn't find any mathematical result on the expression of binomial(b, a) when b < a (which does not mean there is none).

What is also disturbing is than convert(binomial(b, a), factorial) produces an error when b < a.
Which seems to mean that the object convert produces is not the same than the original one expressed differently.

In fact all this leaves me a bit hungry in spite of your excellent work.

@dharr 

I didn't know this tip to avoid the syntax error, very valuable information.
Good luck for the continuation.

@dharr 

Hi,
You missed one: the loop starts with m=2 and thus dff(..., eta$(m-2)) is not correct :-)

@acer 

I suspect the OP might have had something like this in mind

k  := sqrt(sigma^2-`V__+`):
kt := convert(taylor(k, `r__∝` = 100, 30), polynom):
int(kt, `r__∝`)

If you have a true problem to solve, why not use  GraphTheory:-TravelingSalesman instead of ChatGPT?
Unless it was just to see what ChatGPT is capable of?

@Carl Love 

It's a real pity 2D mode is so permissive; such a typing error would be immediately noticed in 1D mode.
Thank goodness  plots:-dis play(...)  produces an error in 2D mode.

Thank you @acer

@acer 

Pretty, I vote up.

Is it possible, as with animate, to change the numer of frames per second when you click on the animation right triangle?

@Carl Love 

That is indeed a good explanation and, although I am still an amateur at Maple, I do believe that this package would benefit from a rewrite.

@Nicole Sharp 

Thanks for the information Nicole.
I really appreciate your last site which brings back old memories.