## Real Analysis: Dyadic Series

Can anyone give me any help on how to get started, or how to do this problem?
---
Prove that if the terms of a sequence decrease monotonically (a_1)>= (a_2)>= ....
and converge to 0 then the series [sum](a_k) converges iff the associated
dyadic series (a_1)+2(a_2)+4(a_4)+8(a_8)+... = [sum](2^k)*(a_2^k) converges.

I call this the block test b/c it groups the terms of the series in blocks
of length 2^(k-1).
----
thank you!

 PhysOrg.com science news on PhysOrg.com >> Front-row seats to climate change>> Attacking MRSA with metals from antibacterial clays>> New formula invented for microscope viewing, substitutes for federally controlled drug
 Recognitions: Homework Help Science Advisor Can you show that this pair of inequalities is true: $$2 \times \sum_{i=1}^{2^k} a_n \geq \sum_{i=0}^{k} \left( \sum_{j=2^{i-1}}^{2^i} a_{2^{i-1}} \right) \geq \sum_{i=1}^{2^k} a_n$$ (The middle expression is the dyadic series.)