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 $\displaystyle a_n=\frac1n$ then $\displaystyle \left(a_n\right)$ and decreasing and converges but $\displaystyle \left(2^na_{2n}\right)=\left(\frac{2^{n-1}}n\right)$ diverges.
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).
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 $\displaystyle \left(a_n\right)$ is a decreasing sequence, isn’t it? Then each odd term has be lie between consecutive even terms.