@acer @dharr

...calculating in the same continuously open worksheet is made again and again ..., the simple result term (6*n-5)/16 no longer occurs; the calculation stops before it. What does this mean?

@dharr 

I would never have thought of the last two simplify commands, thank you very much. The variety of commands created by nesting is impressive.

@vv 

Please show your solution, I would like to try to learn it. Interestingly, simple result terms arise for all even exponents.

@janhardo 

BTW:

One of the typical problems is, for example, the following:
Determine the limit of
cos(pi*sqrt(x^2+x+1)) for x-->infinity.
Can Maple calculate this directly? On the other hand, a verifiable, step-by-step solution is required, one that doesn't deliver the answer immediately at the push of a button. One possible solution ultimately contains the answer to the original question.

Using

"assume(k, integer);
eval(sin(k*Pi));
                               0"

@nm 

It is an ODE with separated variables - almost a mental arithmetic task if you know some antiderivatives.

@janhardo 

The history of this problem can be traced back:
https://books.google.de/books?id=fmRkK_ho2vUC&printsec=frontcover&hl=de&source=gbs_ge_summary_r&cad=0#v=twopage&q&f=true
The theoretical background is also outlined on page 313 in Problem 171. But I was mainly interested in what Maple does with it. Now I have a lot to learn about it :-) It's impressive how the theory, which once seemed so dry, can now be experienced "alive and in color."

Many thanks.

@dharr 

...... is close to the theoretical result 5/2, which can be achieved with a trick using pen and paper.

@Kitonum 

...I had hoped that using gcd(m,n)=1 for the summation an index set of ordered pairs could be generated.

@Carl Love 

"Since the series is alternating, I decided to pair consecutive terms to make it a series of positive terms:" ...is only permitted if convergence is proven. However, this is clearly the case here according to the Leibnitz criterion for alternating series, and you have applied it.

@vv and @Carl Love

Both solutions have been instructive and enjoyable for me.

@janhardo 

...I would like to note:
In this original problem, I am solely interested in how a clever sequence of commands in Maple leads step by step from input to the result. A classic solution using "per pedes ;-)" calculations and the correct solution pi/4+1/2*ln(2) have long been known to me, and this is not the subject of my question. Since I am quite sensitive to convergence issues (especially regarding CAS), I missed some of the individual steps of your solution in your post. This is probably where the misunderstanding stems from—sorry. Therefore, a corresponding addition to your post would be helpful for my Maple learning success.

@janhardo 

......to put the factor j in front of the infinite sum over k?

@vv 

... and thank You.

@vv 

... the Riemann sum I know. As far as I remember, this problem comes from a short book by Williams and Hardy. But I'm interested in getting a result in Maple directly, without much preparation- just a maple exercise for me.