Relation of two complex series

In summary, we can't find a sequence ##\left\{n_k\right\}## such that ##\sum_{n_k+1}^{n_{k+1}}{|a_n|^2}>1##.
  • #1
gustav1139
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0

Homework Statement



Suppose that ##\left\{a_n\right\}## is a sequence of complex numbers with the property that ##\sum{a_n b_n}## converges for every complex sequence ##\left\{b_n\right\}## such that ##\sum{|b_n|^2}<\infty##. Show that ##\sum{|a_n|^2}<\infty##.

Homework Equations


The Attempt at a Solution



We know that ##\lim{a_n}=0##, since if that were not the case then for ##b_n=\frac{1}{n a_n}##, ##\sum{|b_n|^2}<\infty##, but ##\sum{a_n b_n}=\sum{\frac{1}{n}}## diverges. Not sure that's helpful though.My other thought was to try to prove the contrapositive, that given ##\left\{a_n\right\}## such that ##\sum{|a_n|^2}## diverges, we could find a ##\left\{b_n\right\}## such that ##\sum{a_n b_n}## diverges as well.

So we can find a sequence ##\left\{n_k\right\}## such that ##\sum_{n_k+1}^{n_{k+1}}{|a_n|^2}>1##. Then of course, we'd like to pick b's in such a way that ##\sum{a_n b_n}##, while they still converge in the square. But since we don't know how far apart the ##n_k## are, I can't figure out a way to do that. If ##b_n=\left|\frac{\bar{a_n}}{n}\right|##, then the b's converge the way we want them to, but it's not clear that ##\sum{a_n b_n}## diverges. On the other hand, if we choose something that depends on ##a_n## in some way, which seems more promising in some ways, then it's not clear that the b's converge the way they're supposed to. For instance if ##b_n=\left|\frac{\bar{a_n}}{c_n}\right|##, where ##c_n=k## when ##k<n\leq k+1##, then the c's are growing pretty slowly (presumably) so I shouldn't think the b's would converge properly.
 
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  • #2
It seems like it may have to see with Cauchy-Schwarz:

We know if (bn,bn) < oo , then (an,bn)^2<oo . Let me try some more.
 
  • #3
gustav1139 said:
We know that ##\lim{a_n}=0##, since if that were not the case then for ##b_n=\frac{1}{n a_n}##, ##\sum{|b_n|^2}<\infty##, but ##\sum{a_n b_n}=\sum{\frac{1}{n}}## diverges. Not sure that's helpful though.
This argument doesn't work for an=1 since ##\sum{|b_n|^2}## will diverge.
 
  • #4
vela said:
This argument doesn't work for an=1 since ##\sum{|b_n|^2}## will diverge.

...##\sum{\frac{1}{n^2}}## converges... doesn't it?
 
  • #5
D'oh! Never mind. ;)
 

Related to Relation of two complex series

1. What is the definition of a complex series?

A complex series is a sequence of terms that can be written in the form of a+bi, where a and b are real numbers and i is the imaginary unit. The sum of all the terms in the series is called the value of the series.

2. How is the relation between two complex series determined?

The relation between two complex series can be determined by comparing their individual terms and finding a pattern. This can involve calculating the sum or product of the terms, or looking for common factors or differences.

3. Can two complex series be added together?

Yes, two complex series can be added together. This is because the sum of two complex numbers is also a complex number. However, the value of the resulting series may vary depending on the individual terms and their relation.

4. What is the significance of the convergence of two complex series?

The convergence of two complex series is important in determining the validity and accuracy of their relation. If both series converge, it means that their values are finite and the relation between them is valid. If one or both series diverge, the relation may not be accurate or applicable.

5. How can the convergence of two complex series be tested?

The convergence of two complex series can be tested using various methods such as the ratio test, root test, or comparison test. These methods involve comparing the terms of the series to a known sequence or using mathematical formulas to determine if the series converges or diverges.

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