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Find a_n & b_n such that

  • #1

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].
 

Answers and Replies

  • #2
berkeman
Mentor
56,825
6,790

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
34,043
9,891
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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