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I'm trying to solve for n in the following inequality
|4^(1/n) - 1| < x
where n is a natural number and x is a positive real number. I feel like i should be able to do this but i can't think of how. Could someone please help me out with this?
I'm also at a loss for how to solve for n with
100^n / n! < x
but I think i might be able to figure that out once i know how to work with the first one.
There are several difficulties with this problem and the replies to it.
If then does not exist.
That is one difficulty the proposed solution.
On the other hand, if then because .
That means that , another difficulty as pointed out in reply #5.
If then the solutions are
Thanks for the help so far everyone. I've been trying to wrap my head around them for a while now
Regarding when 0 < x < 1, i see how we get to 1 - x < 1 < 4^(1/n) < 1 + x
but when we use ln, don't we get n > ln(4) / ln(x + 1) > (n * ln(1 - x)) / (ln(1 + x) ?
when you used ln you dropped the last part for some reason.