Question

Determine whether the following sequence is increasing, decreasing, or not monotonic. Is the sequence bounded?

Answer 1

(a) a_n = cos(n*pi / 2)
(b) a_n = n/(n^2 + 1)

Answer 2

first one i would just put in n = 1, 2, 3, and see what you get

Answer 3

it is certainly bounded because cosine is bounded above by 1 and below by -1
did you try plugging in the numbers? do 4 and it will be very clear

Answer 4

Oh I see, but what is the part b bounded by?

Answer 5

I know it decreases but I still need to find the bounds.

Answer 6

did you get 0,-1,0,1,0,-1,... for the first one?

Answer 7

For the monotonic sequence theorem, does the sequence have to be bounded on BOTH sides? (top and bottom)?

Answer 8

it is bounded below by 0 for sure

Answer 9

wait are you talking about question 1 or 2?

Answer 10

and yes I got that.... for part a. I got the same pattern.

Answer 11

ok so on to part b

Answer 12

What is part b bounded by? Yes ok

Answer 13

you know the limit is zero right?

Answer 14

yes

Answer 15

That is only the lower bound. Don't we need to find an upper bound also?

Answer 16

are you trying to show it is monotone and bounded?

Answer 17

Yes^

Answer 18

ok so it is bounded above by \(\frac{1}{2}\) but that is not important
it is bounded below by 0 because the terms are all positive
so what is left is to see if it is monotone decreasing

Answer 19

and it does decrease. how did you receive the bounded above part?

Answer 20

also does the sequence have to be bounded ABOVE and BELOW to be counted as bounded?

Answer 21

i replaced n by 1
if it is decreasing it is largest when n is smallest

Answer 22

n could be negative numbers though, correct?

Answer 23

no

Answer 24

????

Answer 25

\(a_n\) n is a positive integer

Answer 26

\[a_1,a_2,a_3,a_4,...\]

Answer 27

otherwise nothing would be anything if you could go to either plus or minus infinity, what a mess that would be

Answer 28

Sounds good, thanks

Answer 29

the easiest way to show the sequence is monotone decreasing is the take the derivative of the corresponding function
\[f(x)=\frac{x}{x^2+1}\] and show that if \(x>1\) the derivative is negative, so the function is decreasing and therefore so is the sequence

Answer 30

otherwise it is very tedious to show it is decreasing, even though it is obvious

Answer 31

I getcha, thanks

Answer 32

yw