Question

Calculus

Answer 1

how can i tell if a sequence is bounded or not?

Answer 2

for example\[a_n= 5n + \frac{ 1 }{ n }\]

Answer 3

If you look at this problem you will quickly see that the sequence is not bounded. This is because every second value is four times as large as the value two before it (4, 16, 64, etc). The same is true of the negative answers. The trick to this problem is to compare a(n) with a(n+2) Try dividing the larger by the smaller

so: a(n+2)/a(n) = (-2)^(n+3)/(-2)^(n+1)

this can be rewritten as [(-2)^(n+1) x (-2)^2]/[(-2)^(n+1)]

You can cancel the (-2)^(n+1) terms which gives you: (-2)^2 = 4.

For all a(n) there exists an a(n+2) that is four times larger (or four times MORE NEGATIVE) as the case may be. As a result, this is not bounded either above, or below.

Answer 4

i am trying to see what you wrote still... and understand it

Answer 5

so, if it goes to infinity, its not bounded...?

Answer 6

ya..

Answer 7

so 5n + 1/n is not bounded because of the inf in the denom?

Answer 8

thanks guys

Answer 9

A sequence \(a_n\)
\(
|a_n| \leq k\) for all \(n \in \mathbb{N}\). That is, for any number in the sequence, there is some finite value k that will always be bigger than that element. Therefore, if:
\[\lim_{n \to \infty} a_n = \pm \infty \]
the sequence must be unbounded. If you take this limit fory ou sequence, you will see this. Please note though, a sequence can still have a an infinite limit that doesnt exist but be still be bounded