n is a naturel number
The question is lacking a bit in definition, (is an integer and is it greater than 1 ?), but it does make sense.
(BTW, I'm still learning LaTex, can anyone tell me how to line up these n's with the rest of the text ?)
Assuming that is an integer greater than 1, it is saying, by way of example, that if
the equation has a root between and It does, 1.618 (3d).
the equation has a root between and It does, 1.839 (3d).
To show that the result is correct in general, define
Then, and showing that will be sufficient.
Without going into the analysis,the second term is always greater than 1, (
(Check it out numerically for and and then use the fact that the numerator is an increasing function, while the denominator tends to e from below).