1. ## decreasing sequence convergence

Suppose that {an} is a decreasing sequence with
lim an =0.

Show that {an} converges if and only if {2^n a(2n)} converges.
(as n goes to infinity)

2. Originally Posted by flash89
Suppose that {an} is a decreasing sequence with
lim an =0.

Show that {an} converges if and only if {2^n a(2n)} converges.
(as n goes to infinity)
Hi flash89.

I think you may have stated your question incorrectly. If $a_n=\frac1n$ then $\left(a_n\right)$ and decreasing and converges but $\left(2^na_{2n}\right)=\left(\frac{2^{n-1}}n\right)$ diverges.

3. Right sorry i meant,
Suppose that {an} is a decreasing sequence with
lim an =0.

Show that the sum of {an} converges if and only if {2^n a(2n)} converges.
(as n goes to infinity)

4. Originally Posted by flash89
Right sorry i meant,
Suppose that {an} is a decreasing sequence with
lim an =0.

Show that the sum of {an} converges if and only if {2^n a(2n)} converges.
(as n goes to infinity)
That still can't be right, as it only places a restriction on the even terms, so the odd terms can be 1/(2k+1) and then the sum diverges whatever the even terms are doing if they are all positive (or if the sum of the even terms converges).

CB

5. Originally Posted by CaptainBlack
That still can't be right, as it only places a restriction on the even terms, so the odd terms can be 1/(2k+1) and then the sum diverges whatever the even terms are doing if they are all positive (or if the sum of the even terms converges).
But $\left(a_n\right)$ is a decreasing sequence, isn’t it? Then each odd term has be lie between consecutive even terms.

6. Originally Posted by TheAbstractionist
But $\left(a_n\right)$ is a decreasing sequence, isn’t it? Then each odd term has be lie between consecutive even terms.
Opps .. missed that.

CB