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Omni-MATH / For every positive integer n, let f(n), g(n) be the minimal positive integers such that 1+(1)/(1!)+(1)/(2!)+…+(1)/(n!)=(f(n))/(g(n)). Determine whether there exists a positive inte…

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problem

For every positive integer nn, let f(n)f(n), g(n)g(n) be the minimal positive integers such that 1+11!+12!+…+1n!=f(n)g(n).1+\frac{1}{1!}+\frac{1}{2!}+\dots +\frac{1}{n!}=\frac{f(n)}{g(n)}. Determine whether there exists a positive integer nn for which g(n)>n0.999ng(n)>n^{0.999n}.
Plain-text mathematical notation (without MathML)
For every positive integer n, let f(n), g(n) be the minimal positive integers such that
1+(1)/(1!)+(1)/(2!)+…+(1)/(n!)=(f(n))/(g(n)).
Determine whether there exists a positive integer n for which g(n)>n^(0.999n).
Original LaTeX notation
For every positive integer $n$, let $f(n)$, $g(n)$ be the minimal positive integers such that
\[1+\frac{1}{1!}+\frac{1}{2!}+\dots +\frac{1}{n!}=\frac{f(n)}{g(n)}.\]
Determine whether there exists a positive integer $n$ for which $g(n)>n^{0.999n}$.

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Source and history

task_id
f29b6c3e-187d-59ac-92ef-cd6b941fa00d
task_key
test--f29b6c3e-187d-59ac-92ef-cd6b941fa00d
task_revision_id
1
config / split
/ test

Official source

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