Series tests containing n!

[oasisinau]

I have a query about n! in various root, ratio etc tests for convergence.

Consider f^n(x^n)=n!
I have that 3 different n’s in this equation are:
- n in f^n as an indicator of order
- n in x^n as the exponent or degree of the variable x
- n! as factorial n, ie any integer

My question is why n! Is not considered to be a constant and (as such) moved to the left of summation in all series tests

Thanks
Owen

 

[Phrzby Phil]

I'm not sure what you're proposing or asking here, so I've waited for anyone else to chime in.
That not having happened, please clarify.

There is no summation in the equation, so I don't get the question.

Are you proposing there is a functional expression for f meeting these criteria?

Taking some values for n:
f(x) = 1
f^2(x^2) = 2
f^3(x^3) = 6
etc.

Is this where you're going?

Thanks.

 

[Phrzby Phil]

I wonder why someone posts a question and then a week later has not responded to an answer.