# monotonic

• Apr 21st 2009, 09:45 PM
needhelp101
monotonic
(e^n)/n

i have to determine if this sequence is monotonic and discuss its boundedness. but i have no clue on how to start this or anything. can someone please help.
• Apr 21st 2009, 10:14 PM
Jameson
A good place to start is to familiarize yourself with what it means to be monotonic. Simply put, if F is monotonic then for any $x \le y \rightarrow f(x) \le f(y)$. So either F will be always growing in value (monotonically increasing) or always declining in value (monotonically decreasing). You might also want to notice that this means that the derivative of F must always be the same sign.
• Apr 21st 2009, 11:54 PM
redsoxfan325
Quote:

Originally Posted by needhelp101
$\frac{e^n}{n}$

i have to determine if this sequence is monotonic and discuss its boundedness. but i have no clue on how to start this or anything. can someone please help.

Check the sign of the derivative. Positive means increasing, negative decreasing. If it switches signs then you can only specify intervals on which the sequence is monotonic.

To see whether it's bounded, take the limit as $n\to\infty$ and use L'Hospital's Rule.

Solution for Monotonicity:
Spoiler:
$\frac{d}{dn}\left[\frac{e^n}{n}\right] = \frac{ne^n-e^n}{n^2} = \frac{e^n(n-1)}{n^2}$.

Since $e^n$ and $n^2$ are always positive $f'(n)\geq 0$ for $n\geq 1$. So this sequence is monotonically increasing on $[1,\infty)$.

Solution for Boundedness:
Spoiler:
$\underbrace{\lim_{n\to\infty}\frac{e^n}{n} = \lim_{n\to\infty}\frac{e^n}{1}}_{L'Hospital's~Rule } = \infty$ so this sequence is unbounded.
• Apr 22nd 2009, 12:18 AM
woof
Quote:

Originally Posted by needhelp101
(e^n)/n

i have to determine if this sequence is monotonic and discuss its boundedness. but i have no clue on how to start this or anything. can someone please help.

Monotonicity is about always either increasing or decreasing, and boundedness is a question of whether or not this expression can "get big" or not.

Your intuition should tell you what direction to go in. This is an exponential over a polynomial, and as n gets large, we know that the exponential dominates, "it wins". So as n gets large, this expression will eventually behave like the exponential and get larger and larger without bound.

So......you know that you need to show the limit is infinite, thus the expression is unbounded, and to also show that it increasingly gets larger, you would need to show that the derivative is always positive.

The details of all that are provided by the posters above.