would you like to give some examples with those enhancements?

@Carl Love thank you for your explanation.

I guess there must be some changes from m12 to m17.

that's the reason some points apprear on the figure plotted by m17.

does it means that the Student[Calculus1]:-Roots command is more accurate in m17 than that in m12 because it founds the extra points, though that's not wanted.

@tomleslie thank you for your explanation, I agree.

@Carl Love The new definition of Frem works, it's better than the original frem.

thank you for your help

@Carl Love I'm interested in the method of sol3, which you said that can be handled by using the procedural form of input to dsolve. I tried, but failed. would you like to show me that method?

@Axel Vogt the process of your solution is so tricky,I want to know when the interval is so complicate that can't preprocess as you done, then how to get the symbolic solution with the int command?

@Axel Vogt Thank you.

I'm sorry. I'm wrong. I missed something. 

Both mma and maple give the same results(about 997968).

Thank you very much, my friends.

@Carl Love I know cos(70) = 0.633319, any calculator can give this result, I believe that's correct.

What I want to know is  the exact value of r, the partial sum (only the first three and the last three terms of the cos series).

I give the code in mma and maple, the results are different.

Maybe only one is right or neither are right? I don't know.

Thank you for your attention.

Because what MMA does w.r.t. "working precision" is not open, I doubt the result(0.633319)too.

But how to check the result,  are the two results reliable?

Is there a method to verified the results?

Still need help.

here is the mma code:

Clear["Global`*"]
poly = Normal[Series[Cos[x], {x, 0, 200}]];
N[poly[[{1, 2, 3, -3, -2, -1}]]]

Plot[poly, {x, 0, 100}, PlotRange -> {-2, 2},
PlotStyle -> {Thickness[0.016], Black}]

poly /. x -> 70 // Short

N[poly /. x -> 70]

poly /. x -> 70.

poly /. x -> N[70, 20]

Precision[%]

poly /. x -> N[70, 200]

Precision[%]

Plot[poly, {x, 0, 100}, WorkingPrecision -> 200,
PlotRange -> {-1.3, 2}]

@Kitonum thank you for your help,

I forgot to give my result, in maple the value of r is about 9.98e5, the partial sum (only the first three and the last three terms of the series) is higher than 1.

I doubt the result, not because it's not same with the result of mma(0.633319), but the results are different.

It puzzeled me. I know the output of evalf[6](add((-1)^(n-1)*70^(2*n-2)/(2*n-2)!, n=1..122)); is the same with mma.

Then I want to know what's the exact value of r(only the first three and the last three terms of the series).

@Preben Alsholm thank you for your reply,

I copied your code and run it in maple, the value of r is about 9.98e5, the partial sum (only the first three and the last three terms of the series) is higher than 1.

Is it correct? I don't know.

@Carl Love Thank you for your reply.

I will take a look in detail.

thank you very much.

@Kitonum thank you for your reply.

The adaptive function with implicitplot make me a little disappointed.

If I have not use the fsolve first,  maybe I guess that system of equations don't have roots.

The adaptive function should be improved.

@Carl Love The blue line is plot well use 1000 points, but the red line use 10000 points?

I wonder why not make auto adaption plot work for this.

Or is there some adaptive option work for this? 

the vector field plot in maple in so terrible, both 3d&2d.

is there a plan to improve it?