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Sum Exponent Ratio Evaluation (Posted on 2023-05-27) Difficulty: 3 of 5
        n + n2 + n3 +...  ...+ nn
Limit  --------------------------
nā†’āˆž    1n + 2n + 3n +...  ...+ nn

See The Solution Submitted by K Sengupta    
Rating: 5.0000 (1 votes)

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Some Thoughts numeric approximation | Comment 1 of 2
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
def f(n):
    for i in range(1,n+1):
        num += n**i
        den += i**n
    return num/den

for n in range(1,20):

  Posted by Larry on 2023-05-27 10:50:01
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