@golnaz Maple 2017 returns an answer without problem. Please check your input again, it has no squareroot.

@one pound , you may have a look at Wikipedia as well, https://en.wikipedia.org/wiki/Logarithmic_integral_function

@brian bovril 

You neither give your file nor say something about your E-Mail client. There is no reason to assume that the client will gather all what is needed. May be you want to pack it in a zip file, respecting the file structure (and you may wish to check whether Maple uses relative or absolute references in the exports).

It is barely readable (I have not checked any of the syntax), but "looking fast enough": the output is overwritten (at least), so perhaps you need a line feed.

@tsunamiBTP 

Using a method without derivatives (like bracketing) "immediately" solves dS2 = 0 (though evaluating the 500 terms is a bit slow except you switch to evalhf or use a compiled version). I am not aware that Maple provides that as an option.

Strange. You may try dS2:=diff(S2, x) and fsolve(evalf(dS2), x=0 .. 1/200, maxsols=1).

@acer Thank you for the fine summary, I like it.

Win 7 64 bit , Maple 2017.3 32 bit, classical interface and using my standard Digits:=15 works,
after ~ 1 minute it returns -.119716579187441-.207411617787594*I

@_Maxim_  ok:

223336.mws
 

Download 223336.mws

@_Maxim_  for example:

Int(sin(t)^(1/3), t = 0 .. Pi/2) + Int(arcsin(t^3), t = 0 .. 1);
op(1,%) + Change(op(2,%), arcsin(t^3)=tt, tt): subs(tt=t, %):
combine(%); # needed
value(%);

                                  Pi
                                 ----
                                  2

@vv do you mean this (attached pdf)? IMG_0002.pdf

@vv another way often works for errors: replace constants by variables, here 1/2 = a and Maple will do

@Markiyan Hirnyk why do you not just stop to insult members?

I do not quite understand the very question (computing -log(1-z) = the formal answer in that way is known to be the wrong way in general, even with series acceleration).

@Markiyan Hirnyk, the topic is finished for me.