Evaluate:
n + n^{2} + n^{3} +... ...+ n^{n}
Limit 
nāā 1^{n} + 2^{n} + 3^{n} +... ...+ n^{n}
From my previous comment:
The numerator should be:
n * (n^n  1)/(n1)
Evaluating the denominator for different values of n leads to Sloane's oeis A031971.
And looking through the notes, there are several approximations for the value of this series.
One is: a(n) is asymptotic to (e/(e  1))*n^n.  Benoit Cloitre, Dec 17 2003
So taking the numerator as: n * (n^n  1)/(n1)
and taking the denominator as: (e/(e  1))*n^n
Cancel out n/(n1)
Cancel out (n^n  1)/n^n
And we have
(e  1)/e =~ 0.632120558828558
Which is close to my f(10,000) = 0.6322235834032078

Posted by Larry
on 20230527 14:35:27 