Finding an oscillating sequence that diverges and whose limit is zero.

[4zimuth]

Homework Statement

Hi, I need to find an oscillating sequence whose limit of the differences as n approaches infinity is zero but the sequence itself is diverging.

Homework Equations

None.

The Attempt at a Solution

My initial guess was:

$$\frac{sin(ln(n))}{ln(n)}$$

 

[HallsofIvy]

What do you mean by an "oscillating" sequence?

 

[mihajovics]

Hi!

If I understand correctly, then
$$sin\left(\sqrt{n}\right)$$
is what you are looking for. The proof is based on the idea sin(a)-sin(b) < a-b and sqrt(n+1)-sqrt(n) converges to 0.
I originally bumped this post, because I need help with a similar problem, where not
a_n+1-a_n converges to 0, but a_n+1/a_n converges to 1.
Does anybody know a good example for the latter? So the problem again:
We need a sequence, that:
1) oscillates (oscillation is when it divergates, but neither to infinity nor negative infinity, eg. -1 +1 -1 +1 ... ; +1 -2 +3 -4 +5 -6 ... ;sin(n) ; etc.)
2) a_n+1/a_n converges to 1