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

1. Mar 10, 2008

### 4zimuth

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

None.

3. The attempt at a solution

My initial guess was:

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

2. Mar 10, 2008

### HallsofIvy

Staff Emeritus
What do you mean by an "oscillating" sequence?

3. Oct 5, 2009

### 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
an+1-an converges to 0, but an+1/an 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) an+1/an converges to 1