Show is convergent iff. , and that

(i) assume is convergent.

Because it is convergent, there is some value alpha such that:

I am not sure if I am on the right track, and if I am, what should I do next?

Printable View

- May 26th 2011, 11:13 AMdwsmithConvergence
Show is convergent iff. , and that

(i) assume is convergent.

Because it is convergent, there is some value alpha such that:

I am not sure if I am on the right track, and if I am, what should I do next? - May 26th 2011, 11:26 AMTheEmptySet
For the forward implication I would break the proof into two parts.

First show that

I assume that you mean the above if k=0 the sequence is constant and equal to 1.

So suppose not( assume k < 0) and use what you have above to get a contradiction. Hint the sequence will be unbounded.

Then after you know that k > 0 you can make a direct argument to show that it converges to zero.

The reverse <= implication is not bad at all. - May 26th 2011, 11:31 AMTinyboss
Instead of doing "k non-negative implies convergent" and "convergent implies k non-negative", it'll probably be easier to do "k non-negative implies convergent" and "k negative implies divergent".

There are probably nicer ways to do it, but for k non-negative, I'd choose m such that 1/m<k. Then , and since you know as , so must , and so . The case where k is negative follows immediately. - May 26th 2011, 11:54 AMdwsmith
(a) by contradiction:

is convergent and k < 0.

Since k < 0, we can right

Thus, we have reached contradiction and the sequence is convergent for - May 26th 2011, 12:00 PMdwsmith
(b)

Good? - May 26th 2011, 12:15 PMTheEmptySet
Since this is for an analysis class I would be a bit more explicit.

Since we know that

This implies that for every epsilon greater than 0 there exists an N such that ... so set

But this implies that the sequence is bounded. But since k < 0 we can make

as large as we wish.

Let M > 0 be any real number

Then Pick

Then for any n > n'

so we have the desired contradiction. - May 26th 2011, 02:19 PMdwsmith
- May 27th 2011, 08:52 AMTheEmptySet
Lets look at a concreate example.

Let

and set

Now notice if I set

Now for any

if

This shows that if k is negative the sequced can be made as large as we want. There is nothing special about one million the above is true for any positive M so the sequence diverges to infintity.