Series question (non-baby rudin, ch. 4, #7)

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In summary, the conversation discusses a proof for the convergence of a sequence of positive numbers given certain conditions. The solution involves choosing the c_k values to ensure that one sum is finite while the other is infinite, using the Cauchy-Schwarz inequality.
  • #1
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Homework Statement



Suppose [tex]\{a_n\}[/tex] is a sequence of positive numbers such that [tex]\sum_na_nb_n < \infty[/tex] whenever [tex]b_n \ge 0[/tex] and [tex]\sum_nb_n^2 < \infty[/tex]. Prove that [tex]\sum_na_n^2 < \infty[/tex].

Homework Equations



There's a suggestion in the text:
If [tex]\sum_na_n^2 = \infty[/tex], then there are disjoint sets [tex]E_k \, (k=1,2,3,\dots)[/tex] so that
[tex]\[\sum_{n\in E_k}a_n^2 > 1.\][/tex]
Define [tex]b_n[/tex] so that [tex]b_n = c_ka_n[/tex] for [tex]n \in E_k[/tex]. For suitably chosen [tex]c_k[/tex], [tex]\sum_na_nb_n = \infty[/tex] although [tex]\sum_nb_n^2 < \infty.[/tex]



The Attempt at a Solution


Using the hint, we've got
[tex]
\[b_n^2 = c_k^2a_n^2; \qquad\qquad a_nb_n = c_ka_n^2\]
\begin{align*}
S1 = \sum_n b_n^2 &= \sum_{n\in E_1} c_1^2 a_n^2 + \sum_{n\in E_2} c_2^2 a_n^2 + \dots \\
&= \sum_k \sum_{n_k\in E_k} c_k^2 a_{n_k}^2 < \infty\\
S2 = \sum_n a_n b_n &= \sum_k \sum_{n_k\in E_k} c_k a_{n_k}^2 = \infty.
\end{align*}
[/tex]

From this, it seems clear that we want to choose the [tex]c_k[/tex] to be "sufficiently dampening" so that [tex]S1[/tex] really is finite but [tex]S2[/tex] is not. I'm not entirely sure how to go about choosing the [tex]c_k[/tex] for arbitrary [tex]a_n[/tex] and [tex]b_n[/tex]. Thanks very much, any hints would be greatly appreciated! Sorry about the formatting, I'm new here..
 
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  • #2
Perhaps a well-known inequality dealing with sums would help.
 
  • #3
Thanks for the hint! I'm assuming you're referring to the CBS inequality, but I'm not quite sure how to apply it in a useful way.

I've got
[tex]
\sum_n a_nb_n \le \sqrt{\sum_n a_n^2}\sqrt{\sum_n b_n^2}
[/tex]
and the RHS is infinite since the sum of the squares of the a_n is infinite.
 

Related to Series question (non-baby rudin, ch. 4, #7)

What is a "series question" in the context of non-baby Rudin, chapter 4, problem #7?

A "series question" in this context refers to a mathematical problem involving the convergence of a series of numbers.

What is the purpose of studying series questions?

Studying series questions allows for a deeper understanding of mathematical concepts such as convergence, divergence, and the behavior of infinite sequences.

What is the difference between a convergent and a divergent series?

A convergent series is one in which the sum of its terms approaches a finite limit, while a divergent series is one in which the sum of its terms increases without bound.

How can I determine if a series is convergent or divergent?

There are various tests and techniques that can be used to determine the convergence or divergence of a series, such as the ratio test, the root test, and the comparison test.

What real-world applications are there for understanding series questions?

Series questions have many applications in fields such as physics, engineering, and finance, where the study of infinite sequences and their convergence can help in solving real-world problems.

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