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Is $n^{1/\log \log n} = O(1)$?
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Problem
Is $n^{1/\log \log n} = O(1)$ ?
Suppose that $n^{1/\log \log n} = c$ where $c$ is constant.
Taking logs of both sides,
$$\frac{1}{\log \log n}\log n = \log c.$$
I am not able to spot an error. Please help
Suppose that $n^{1/\log \log n} = c$ where $c$ is constant.
Taking logs of both sides,
$$\frac{1}{\log \log n}\log n = \log c.$$
I am not able to spot an error. Please help
Solution
The function $n^{1/\log \log n}$ tends to infinity, since
$$
n^{1/\log\log n} = e^{\log n/\log\log n},
$$
and $\log n/\log \log n \longrightarrow \infty$.
$$
n^{1/\log\log n} = e^{\log n/\log\log n},
$$
and $\log n/\log \log n \longrightarrow \infty$.
Context
StackExchange Computer Science Q#117613, answer score: 7
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