Prove convergence of series equals convergence of dyadic series

[emilya]

Can anyone give me any help on how to get started, or how to do this problem?
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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).
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thank you!

 

[NateTG]

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.)