Evaluate:
n + n2 + n3 +... ...+ nn
Limit --------------------------
nāā 1n + 2n + 3n +... ...+ nn
I was hoping a numeric approximation would help identify the solution but I couldn't relate what I found with a closed form solution.
I found f(10,000) to be 0.6322235834032078
This was slightly smaller than f(1,000), , suggesting convergence.
The numerator should be:
n * (n^n - 1)/(n-1)
The denominator should be a polynomial which is one degree more than the numerator, suggesting convergence.
That's as far as I got
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def f(n):
num=0
den=0
for i in range(1,n+1):
num += n**i
den += i**n
return num/den
for n in range(1,20):
print(f(n))
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Posted by Larry
on 2023-05-27 10:50:01 |
numeric approximation
