# Monotone sequence - how can this be a monotone sequence if it bounces between numbers

• May 15th 2011, 01:42 AM
iva
Monotone sequence - how can this be a monotone sequence if it bounces between numbers
My understanding of showing a sequence is monotone is to show that it either increases or decreases by working out

\$\displaystyle (a_n+1) - (a_n)\$

and if it's > 0 its increasing, if less than, decreasing, right?

But now i have this problem that I must say if the sequence is monotone:

\$\displaystyle (a_n) = (-1)^n + 2n\$

which to me works out that it is bounded by 0 and 4, ie that it bounces between 0 and 4 ( if n is odd then the value is 0 if even then its 4)

Surely this goes against monotone sequence??? But my answer says that it IS monotone. How can it be monotone when it is neither increasing nor decreasing??
• May 15th 2011, 01:51 AM
Mondreus
Quote:

A monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing.
You have a monotonically increasing sequence since \$\displaystyle a_{n+1}-a_n \geq 0\$, but it's not strictly monitonically increasing.
• May 15th 2011, 01:55 AM
iva
Thanks, I was just replying as I picked up my error (I was confusing the \$\displaystyle a_n + 1 - a_n\$ with the original sequence. Ok I see it is increasing.

But what do you mean by "it is not strictly monotonically increasing?"

Thank you
• May 15th 2011, 02:13 AM
Mondreus
In order for it to be strictly monitonically increasing, you need \$\displaystyle a_{n+1} - a_n > 0\$