Is there a connection between of a function with his:   derative, integral and serie ?

@Carl Love 

Thanks

Yes, i understand it that you mention it for completeness.

Complex analyse is not that easy as it is for real numbers.

@acer 

Thanks, indeed this question has nothing to do with complex numbers and know nothing about complex functions

 

@Carl Love 

Thanks

I do have  some basic understanding of complex numbers, but complex functions( analyse) i did not yet studied.
 

@Carl Love 

Thanks
I did 

convert(ln(1+x),FPS); and get 

 Sum((-1)^k*x^(k + 1)/(k + 1), k = 0 .. infinity)  = 

But value(sum((-1)^k*x^(k + 1)/(k + 1), k = 0 .. infinity)); gives back again 

So f(x)= ln(1+x)   gives a serie, but this serie gives not the function back
Domain is < 0 to ? from ln(1+x)  and serie start at x = 0
It must be functions what their domain ranges from 0 ...n  to get from a serie a function back then?   

Don't know how to handle this vectorvalued function task for 4 points with tangentlines (or velocity lines?) and accelleration lines 

There was a task made  with a parametric curve also with 3 points and tangentlines, but the vectornotation differs
I tried task 8(i) for curve see post

I choose 4 points from t = 0 ,1/4 , 1/2, 3/4  (they must no too close and interesting )

blz35.pdf

blz36.pdf

hfdts2_blz37.pdf

@vv 

Thanks

Is it possible to get a function from a FPS serie ?

Expanding the integrand in a Maclaurin series ( as book tip ) : the integrand is here : ln(1+x).

@janhardo 

I must use this form for the vectorfunction in task 8

s(t):=[t^5-t+1,2*t^6-t^2-t+1,t=-1..1.2], labels=[x,y];

Download info__parametrisch_en_vectorfunctie.mw

@Carl Love 

Thanks

I read here : "formal power series can be seen as an extension of polynomials. We speak of “formal” series, because we are not worried about convergence issues."

But it is still not clear how the proven serie has a connection with the serie expension of this integrand function : how are those two series related ?

Th e proof is not yet fully understood, but maybe later? 

@vv 

Thanks

Can't follow the proof , because i don't know what the intended use for some commands is ( F,Sx and S )

Is it a proof by induction ?

Also the task introduce a integral and suggest to expand the integrand in a serie for getting  also a proof ?

@tomleslie 
Thanks!

I use also help in Maple, yes
Your explanation should also to be read in the manual ?

@tomleslie 

Thanks 

It is all understood now with the graphs, but suppose should i try to program this, then i need the plots of the left-hand-right-hand and mid-sum, that's i what a mean there.
My sentence was not complete, sorry. 

----------------------------------------------------

One command seq has a question : there is a  [ ] included  :  a list-building operator to initialize a list construction ? 

seq
                   ( [ [ X[i],   Y[i+1] ],
                       [ X[i+1], Y[i+1] ]
                     ][ ],
                     i=0..N-1
                   ),
--------------------------------------------

Also filling 1 Array Ymid  is little bit complicated with nested ~  

Ymid:=f~(X+~(b-a)/(2*N)); # X+ 1/2 partition 
indeed powerful with only this ~ symbool to do this

@janhardo 

Amazing programming to see the whole picture.

But using a drawing of the 3 riemann sums is needed (for me ) to see the structure of the points of the tread-riser plot

@tomleslie 

Thanks

Well, i do have some  experience with a simple procedure making and thougth that one from @ CarlLove is a advanced one not for a beginner like me.

But thougth from making a procedure  from you code example seems to me easier to follow.
Yes, i think i can follow it.

Works great with bookexamples!