I applied the following per your suggestions (more appropriately my interpretation of your suggestions):
None of my code for this topic ever did what you've now tried. Instead, my earlier responses to address this problem with memory resources used unapply to create a procedure, so that the only things being added up were floating-point numbers and not any symbolic addends (containing trig calls, etc).
I have what I think you refer to as a procedure for S5 through S10.
You should add up m=10000 floating-point numbers (each computed from evaluating the generic subterm at some specific integer k and some floating-point value for x). As seen above in my Answer, that can even be done fast and leanly under evalhf. But even if you don't want to use evalhf mode, it is far less resource consuming to add up only the individual floating-point values for each k from 1 to m=10000 for each given floating-point x value.
I then inserted S7(100000,0.00002) into the RootFinding:-Analytic command.
My result: Error, (in RootFinding:-Analytic) the function, 0, should be non-constant.
Now if you could point out what you are trying to tell me in the code without condescending remarks that would be good: