Sequences: Monotones, Supermum, Infimum, Min & Max, and Convergence Explained

[mmoadi]

Homework Statement

For the sequence (see picture), explore its monotones, define supermum and infimum, minimum and maximum, and find if the sequence is convergent. If the sequence is convergent, find form where on the terms of this sequence differ from the limit for less than ε = 0.01.

The Attempt at a Solution

a1= 1
a2= 1/4
a3= 1/9
a4= 1/16
a5= 1/25

Concussion: the sequence is decreasing.

Prove:
a(n+1) ≤ an
(1/n+1)^n ≤ (1/n)^n
(1/n+1)^n – (1/n)^n ≤ 0

Since (1/n+1)^n will always be smaller than (1/n)^n, I concluded that the left side will always be smaller than 0. I think this is the prove that the sequence is decreasing.

Now, I stuck. How do you know if the the sequence is monotonic, how can I define supermum and infimum, min and max? And how can I prove if it is convergent or not?

 

[slider142]

From the terms you give, your sequence appears to be (1/n)², not (1/n)ⁿ. The argument you gave in your limit has to be adjusted for this.

 

[mmoadi]

Ooops, my bad. It was a typo. It should be like this:

a1= 1
a2= 1/4
a3= 1/27
a4= 1/256
a5= 1/3125