How Can We Modify Convergent Series to Make Their Product Diverge?

  • Thread starter AfterSunShine
  • Start date
In summary, the problem is to find two series, a_n and b_n, such that both converge, but their product series, when taking the square root of a_n and multiplying it by b_n, diverges. Attempts have been made using a_n = 1/n^2 and b_n = (-1)^n / n and a_n = 1/n^2 and b_n = arctan (1/n), but they have failed. The challenge now is to modify the series in order to keep them converging at a slower rate and find a way to make the product diverge.
  • #1
AfterSunShine
27
3

Homework Statement


Find [tex]a_n[/tex] & [tex]b_n[/tex] such that [tex]\sum_{n=0}^{\infty}a_n[/tex] & [tex]\sum_{n=0}^{\infty}b_n[/tex] are convergent series, but [tex]\displaystyle \sum_{n=0}^{\infty} \left( \sqrt{a_n} \cdot b_n \right)[/tex] diverges.

Homework Equations


None.

The Attempt at a Solution


Try too hard for this but still cannot find such [tex]a_n[/tex] & [tex]b_n[/tex].
 
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  • #2
AfterSunShine said:

Homework Statement


Find [tex]a_n[/tex] & [tex]b_n[/tex] such that [tex]\sum_{n=0}^{\infty}a_n[/tex] & [tex]\sum_{n=0}^{\infty}b_n[/tex] are convergent series, but [tex]\displaystyle \sum_{n=0}^{\infty} \left( \sqrt{a_n} \cdot b_n \right)[/tex] diverges.

Homework Equations


None.

The Attempt at a Solution


Try too hard for this but still cannot find such [tex]a_n[/tex] & [tex]b_n[/tex].

That is not an acceptable post. You must show your efforts before we can offer tutorial help. Show us what you have tried so far please...
 
  • #3
There is nothing to show here
basically I tried a_n = 1/n^2 & b_n = (-1)^n / n but failed
tried a_n = 1/n^2 & b_n = arctan (1/n) but failed
and so on...
 
  • #4
AfterSunShine said:
basically I tried a_n = 1/n^2 & b_n = (-1)^n / n but failed
That is not so bad as a start. Can you modify the series in order to keep them converging, but doing so significantly slower?
And then you'll need some trick to get the product diverging - something that changes the sign flip thing...
 
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Likes berkeman

1. What is the purpose of finding a_n & b_n in a scientific study?

The variables a_n and b_n are often used to represent unknown coefficients or parameters in a mathematical model. By finding their values, scientists can better understand and predict the behavior of a system or phenomenon.

2. What techniques or methods are commonly used to find a_n & b_n?

The techniques used to find a_n and b_n vary depending on the specific problem being studied. Some common methods include regression analysis, numerical optimization, and curve fitting. Scientists may also use computer simulations or mathematical models to determine the values of these variables.

3. How do the values of a_n & b_n affect the outcome of a scientific study?

The values of a_n and b_n can greatly impact the results of a scientific study. These variables often represent important factors that influence the behavior of a system or phenomenon. By accurately determining their values, scientists can make more accurate predictions and draw more meaningful conclusions from their research.

4. Can a_n & b_n be negative or complex numbers?

Yes, a_n and b_n can take on any real or complex values depending on the context of the study. In some cases, negative or complex values may be necessary to accurately represent the behavior of a system or phenomenon.

5. What are the limitations of using a_n & b_n in scientific research?

While a_n and b_n can be useful variables in many scientific studies, they also have limitations. These values may only be applicable in certain conditions or may not fully capture all aspects of a complex system. Additionally, the process of finding a_n and b_n may be subject to error, which can impact the accuracy of the study's results.

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