I tried your example above on my laptop at home, where I am running Maple 9.52.  The line with chebpade gave me the following error:

chp:=chebpade(f(x), x, [5,5]);
Error, (in numapprox:-chebpade) singularity in or near interval

I'm wondering if this is the cause of the difference that you are observing between the two approximations.

>restart: with(plots):

Dummy transfer function below.

>f:=s->(1/(s^2+2*s+1));

Data ranging from 1 to 10 evaluated with function above.

>Data1:=[seq([i,f(i)],i=1..10)];

>plot(Data);

Same as above, overlayed with a continuous plot of the function.

>p1:=plot(f,1..10,color=blue):
>p2:=plot(Data1):
>display(p1,p2);

Maple 11 Classic Interface:

Go to File and select Preferences...

Select the Numerics tab.

Click the Round to box.  Type in a number.  The default is 10.  Apply to Session.  Then re-execute your code to see if this is what you are lookin for.

For Maple 11 Standard Interface, the above can be found under the Tools - Options..Precision tab.

I think the reference above is to the following, from the Maple Applications Center.

www.maplesoft.com/Applications/app_center_view.aspx 

It works under Maple 9.5 and Maple 11 Classic Interface.

>assume(x<>0);

>about(x);
 

Try the following and see if the answer is acceptable to you.  The ~'s tell you that your variables are assumed.

>restart: assume(0 <= s, 0 <= w, 0 <= t, 0 <= h):

>ans:=diff(((w-s)^(1-t)-1)/(1-t)+(((1+r)*s)^(1-t)-1)/((1+h)*(1-t)), s);

>expand(ans);

>solve(%,s);

Thank you for the example.  I will use it for other "test" cases and try to expand upon it. 

It seems recurrences come close for this particular example.  I will keep looking into the "gfun" package.  I came across "Guess" as well but could not manage to make it work.  Thank you for your effort.

Take a look at the help file for the plot command.

?plot/options

>plot(x^2, x=-3..3, labels=["x axis", "y axis"], labeldirections=[horizontal, horizontal]);

The labeldirections can be horizontal or vertical

>plot(x^2, x=-3..3, labels=["x axis", "y axis"], labeldirections=[vertical, horizontal]);

This implies re-plotting.  In Maple 11 there are options if you right click on the plot, but I do not know if you can alter the direction of the axis labels.

>b:=t->piecewise(0>=t,-t,5>t,0,5<=t,t-5); 

>maximize(b(t),t=0..3,location=true);

yields

0, {[{t = 3/2}, 0], [{t = 0}, 0], [{t = 3}, 0]}

What do you expect the answer to be for t=0..3?  Try plotting it and then see if the answers you are getting match your expectations.

>restart:

>b:=t->piecewise(0>=t,-t,5>t,0,5<=t,t-5);

>plot(b(t),t=-2..6,discont=true);

There is an update to Maple 9.5 at:

www.maplesoft.com/support/maple95/General/2.aspx 

I do not have 9.5 loaded on my machine, so I can not test it.  Just a thought.

I posted twice.  This one needs to be removed.

It's Sunday, so maybe I'm not firing on all cylinders, but I'm struggling to understand your question.  Are you refering to the "binomial" command, or to something else.  Where did you see the syntax that you refer to with the big "O".  With respect to the title of your posting, largeness of what?  Sorry if I'm being dense.    

The nice thing about Maple is that you can write your own functions if you can not find them built in.  Maple 11 provides the CumulativeSum function, which is part of the Statistics package.  Maple 9 does not have this, but it very simple to do.  

Maple11:

>restart:

>X:=[1,2,3,4,5,6,7,8,9,10];

>Statistics[CumulativeSum](X);

gives this as a result:

[1.,3.,6.,10.,15.,21.,28.,36.,45.,55.]

Maple 9:

>restart:

>X:=[1,2,3,4,5,6,7,8,9,10];

>v:=array(1..nops(X)):

>for i from 1 to nops(X) do
v[i]:=(add(i,i=1..i))
end do:print(v);

gives this as a result:

[1, 3, 6, 10, 15, 21, 28, 36, 45, 55]