Understanding the Block Test for Convergence of Dyadic Series

In summary, the conversation discusses the proof of the convergence of a series where the terms of the sequence decrease monotonically and converge to 0. The block test is introduced as a way to group the terms in blocks of length 2^(k-1). The conversation also requests help in proving a pair of inequalities related to the dyadic series.
  • #1
emilya
1
0
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!
 
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  • #2
Can you show that this pair of inequalities is true:

[tex]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[/tex]

(The middle expression is the dyadic series.)
 
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Related to Understanding the Block Test for Convergence of Dyadic Series

1. What is a dyadic series in real analysis?

A dyadic series in real analysis is a series of terms that are powers of two, such as 1/2, 1/4, 1/8, etc. These series are commonly used in mathematical analysis to study the convergence of functions.

2. How do you determine the convergence of a dyadic series?

The convergence of a dyadic series can be determined by using the Cauchy convergence test. This test states that a series converges if and only if the sum of the absolute values of the terms in the series is less than or equal to a constant for all positive integers.

3. What is the difference between dyadic series and geometric series?

A geometric series is a series of terms that are multiplied by a common ratio, while a dyadic series is a series of terms that are powers of two. While both series have similar properties, such as being convergent or divergent, they are not the same.

4. Can a dyadic series have negative terms?

No, a dyadic series cannot have negative terms. This is because the terms in a dyadic series are powers of two, which are always positive. However, a dyadic series can have terms that tend to negative infinity, making the series divergent.

5. How is a dyadic series used in real analysis?

Dyadic series are commonly used in real analysis to study the convergence of functions. They are also used in other areas of mathematics, such as probability theory and signal processing. Additionally, dyadic series can be used to approximate functions and solve problems in numerical analysis.

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