# decreasing sequence convergence

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• June 4th 2009, 01:09 AM
flash89
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)
• June 4th 2009, 01:34 AM
TheAbstractionist
Quote:

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.
• June 4th 2009, 01:41 AM
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)
• June 4th 2009, 04:17 AM
CaptainBlack
Quote:

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
• June 4th 2009, 06:05 AM
TheAbstractionist
Quote:

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.
• June 4th 2009, 06:54 AM
CaptainBlack
Quote:

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