# Sequence and bounds (2)

• Mar 8th 2008, 12:06 PM
akhayoon
Sequence and bounds (2)
can some one prove to me why

$\displaystyle (1.2)^{-n}$ is a decreasing sequence?

I know it's obvious but in my textbook they prove it by finding the derivative

which should be $\displaystyle \frac{-(1.2)^{-n}}{ln1.2}$ which is increasing though right?
• Mar 8th 2008, 12:19 PM
Thomas154321
Surely the derivative is just $\displaystyle (-n)(1.2)^{-n-1}$? Which is negative for $\displaystyle n>0$, hence decreasing? Unless I'm misreading the question as I've no clue how logs appeared...

But your derivative is also always negative; either way it's decreasing.
• Mar 8th 2008, 12:23 PM
akhayoon
I don't think u could do that...or I'm pretty sure...because the variable is a power not a base
• Mar 8th 2008, 12:30 PM
Thomas154321
Oh right that was stupid of me, sorry! I redid it and got the same as you. log(1.2) is positive and $\displaystyle (1.2)^n$ is also positive, so you definitely have a negative answer which shows decreasing. I don't see the problem.
• Mar 8th 2008, 12:33 PM
akhayoon
no but $\displaystyle (1.2)^{-n}$ gets thrown to the denominator right? and theres already a negative sign in front of it
doesn't that make it increase to 0?
• Mar 8th 2008, 12:38 PM
Thomas154321
Yes as n tends to infinity the derivative tends to 0, but it is never actually 0, it just gets closer and closer. Any value of n you choose will give a negative answer.
• Mar 9th 2008, 11:22 AM
CaptainBlack
Quote:

Originally Posted by akhayoon
can some one prove to me why

$\displaystyle (1.2)^{-n}$ is a decreasing sequence?

I know it's obvious but in my textbook they prove it by finding the derivative

which should be $\displaystyle \frac{-(1.2)^{-n}}{ln1.2}$ which is increasing though right?

Your sequence is obtained by sampling $\displaystyle f(x)=1.2^{-x}$ this function is decreasing because its derivative is negative. Because the function is decreasing the sequence obtained by sampling it at $\displaystyle 1, 2, ..$ is also decreasing.

(you cannot differentiate a sequence because the index is a discrete variable and you need a continuous variable to differentiate)

RonL