A Question Of Convergence
Does the following converge?
n → ∞ lim n e 2 ! e 2 − 0 . 5 ln 2 + n e 3 ! e 3 − 0 . 5 ln 3 + n e 4 ! e 4 − 0 . 5 ln 4 + …
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1 solution
You are having heavy amount of knowledge truly
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I honestly just guessed it (correct).
It's easy to guess, but not to prove. This question was actually inspired by Michael Mendrin in his response to whether or not 1 + 2 + 3 + … converges.
But I think for any finite n the nested radical itself is inf! For any finite n think how you define that expression. The value of that expression should at least as big as e m ! / n m for all m ∈ N which clearly diverge to infinity as m goes to infinity
The link to Herschfeld's Theorem states the result for a fixed power (root) p. The exponents in the problem are decreasing with the index.
I feel the .... in this problem is a tad ambiguous. I presume there are n terms, but that could be clearer.
By Stirling's approximation , we can see that
x n p n = ( e n ! e n − 0 . 5 ln n ) n n 1 = e ( n / e ) n n n ! → e 2 π
is bounded; hence, the infinitely nested radicals converge due to Herschfeld's Convergence Theorem .