whatlifeforme said:Homework Statement
Determine if convergent or divergent. Determine limit if convergent.Homework Equations
[itex]a_{n} = \frac{n!}{n^n}[/itex]The Attempt at a Solution
As per the hint, i use 1/n to compare.
however, how is this statement true:
[itex]\lim_{x\to\infty} \frac{n!}{n^n} <= \lim_{x\to\infty} \frac{1}{n} [/itex]??
does the denominator become larger faster than the numerator making it a smaller number than the 1/n?
dick said:think about it. E.g. 3!/3^3 is less than 1/3. Why is that? That's (1*2*3)/(3*3*3).
Yes, the denominator becoming larger faster will result in a smaller overall value for the fraction.
The faster the denominator increases, the smaller the overall value of the fraction will be.
Yes, the numerator and denominator can increase at different rates, but the overall value of the fraction will still be affected by the rate of increase of the denominator.
Yes, the denominator can increase at a slower rate than the numerator, resulting in a larger overall value for the fraction.
No, the denominator can continue to increase without limit, resulting in a smaller and smaller overall value for the fraction.