@Carl Love 

The user has m(R+x) (missing `*`). The main problem is using c[1],...,c[4]  and c  followed by  c := ...

@tsunamiBTP 
Yes.
If you just want sinc ant not the DynamicSystems package then define e.g.

sinc := x -> piecewise(x=0, 1, sin(x)/x);

@_Maxim_ 

I totally agree that an epsilon parameter would be nice, but not essential. Note that the epsilon in evalf/Int was also absent in the past. The main problem is the lack of methods.

BTW, the eulermac implementation is far from perfect, see here.

@_Maxim_ 

I don't think that such options are really necessary. Why use an epsilon when Digits should be enough? evalf/Sum usually increases internally Digits.

In your example Maple probably does not have a suitable convergence accelerator.
[The acceleration is usually more efficient for alternating series].

In such cases I think that a combination of symbolic+numeric methods should be used; of course here the symbolic one is sufficient, but take e.g.
evalf(Sum((1-10^(-3))^k/(k+ln(k)), k = 1 .. infinity));

The number of beginning terms is also superfluous because one may use
add(f(n),n=1..N) + Sum(f(n),n=N+1..infinity).

@jacobBN 

With uses in a procedure, the package is not loaded in memory; instead, the necessary functions are invoked by their long names.
with(packagename) can be used only at top-level (i.e. not inside a procedure).

@Markiyan Hirnyk 

Have you read the question or the answer?

@Robert Israel 

Unfortunately there are much simpler expressions for which testeq fails, e.g.

sin((3/7)*Pi)-sin((1/7)*Pi)+sin((2/7)*Pi)-sqrt(7)/2;

@_Maxim_ 

Unfortunately; probably branch problem due to the complex arguments of the elliptics.
MultiSeries seems to be still experimental, so maybe it will be corrected.
 

@_Maxim_ 

From the very beginning:

S1 := int(s(tau),tau=a..t):
E1 := r(t) - diff(r(t),t)*~(S1/s(t)):
MultiSeries:-asympt~(E1,t);

Fine analysis!

I just want to add that the asymptotic behavior of the involute (t-->oo) can be obtained directly using
MultiSeries:-asympt
(for t-->-oo  one has to change t to -t). Of course, MultiSeries:-series(...,t=infinity) is also possible.

@Markiyan Hirnyk 

What should be in your opinion the Maple's answer for your test problem

pdsolve([diff(u(t, x), t, t) = diff(u(t, x), x, x), u(t, 0) = 0, u(t, Pi) = 0]);

Same question in an ideal CAS.

@David1 

I am afraid that Groebner wil not help you very much for this problem. The degree of the polynomials would be huge and also the coefficients.
But, do you really need all the solutions?  You may add constraints to select some of them and then use the Direct Search package https://www.maplesoft.com/applications/view.aspx?SID=87637

It's a formal result.
It is easy to give sufficient conditions for the coefficients to have standard solutions. No CAS will do this in the near future.
Anyway, it is much better than nothing.

P.S. There are plenty of real bugs in Maple. Why not focus on these?

@Markiyan Hirnyk 

This is not really a trick. And it is hard to speak about bugs in this context, see http://mathworld.wolfram.com/ConstantProblem.html