Consider the sequence of divergent series in part evaluated by the following maple input.


f1 := seq((1-a)*(1/2)+sum((-1)^n*(n^(1/n)-a), n = 1 .. infinity), a = 1/10 .. 9*(1/10), 1/10): evalf(f1);




f2 := `$`((1-a)*(1/2)+sum((-1)^n*(n^(1/n)-a), n = 1 .. infinity), a = 2 .. 10): evalf(f2);

The Maple output, which is the MRB constant for each sum, is questionable; because, for a not =1, by the limit test, g(n)= sum((-1)^n*(n^(1/n)-a), n = 1 .. infinity) does not converge because limit((-1)^n*(n^(1/n)-a), n=infinity) DNE.






In the following comments, however, I would  like to discuss the second term of the formula in each of the above divergent series, g(n)= sum((-1)^n*(n^(1/n)-a), n = 1 .. infinity) where a is not=1. In particular I would like to find what value for a gives a minimum absolute value for g where abs(a)<1. I call that value for a the MRB2, after the MRB constant.


I do wonder what Maple's sum command is doing when it returns only a single value for the divergent series. Do you have any idea as to what Maple means to be saying here? Would you like to help me find that value for a?


Please Wait...