Monday, May 9, 2011

A quick exercise on divergent sequences

I have a sequence and I'm trying to show that it converges.  Here's my attempt to turn a morning of frustration into a blog post.

Let s(j), j = 1, 2, ..., be a sequence of real numbers with the following properties:
  • There exist constants a and b such that a <= s(j) <= b for all j.
  • In the limit as j goes to infinity, s(j) - s(j-1) = 0.
Conjecture: Any sequence s(j) satisfying these properties is convergent.

Your job: Disprove the conjecture by providing a counterexample.

There are many possible answers, but I give one in the comments.

1 comment:

Andrew said...

s(j) = sin(log(j))

This sequence satisfies both conditions: -1 <= sin(log(j)) <= 1, and using trig identities it is not too hard to show the second condition. However, it clearly has no limit, as it never stops bouncing between -1 and 1 (though each bounce is on an ever-increasing "period").