Series Absolute Convergence Proof

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SUMMARY

The discussion centers on proving the absolute convergence of the series \(\sum a_n b_n\) given that both \(\sum a_n^2\) and \(\sum b_n^2\) converge. Participants suggest utilizing the inequality \(2|ab| < a^2 + b^2\) as a foundational step in the proof. Additionally, the inequality \((|a| - |b|)^2 \geq 0\) is recommended as a method to further the argument. The conversation highlights the importance of these inequalities in establishing the necessary conditions for absolute convergence.

PREREQUISITES
  • Understanding of series convergence, specifically absolute convergence.
  • Familiarity with inequalities such as the Cauchy-Schwarz inequality.
  • Knowledge of basic calculus and real analysis concepts.
  • Proficiency in manipulating algebraic expressions and inequalities.
NEXT STEPS
  • Study the Cauchy-Schwarz inequality in depth.
  • Explore proofs of absolute convergence for series.
  • Learn about the implications of convergence in real analysis.
  • Investigate additional inequalities useful in series convergence proofs.
USEFUL FOR

Students of mathematics, particularly those studying real analysis, and anyone involved in advanced calculus or series convergence proofs.

harrietstowe
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Homework Statement


If \suman2 and \sumbn2 converge show that \sumanbn is absolutely convergent


Homework Equations





The Attempt at a Solution


I think I should do something with the statement 2ab\leq a^2 + b^2
 
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Yes, you should. Better yet, use 2|ab|<a^2+b^2. Can you prove that?
 
Hey sorry to return to this late but I am still pretty stuck on this. Dick, I don't think I know how to prove that
 
harrietstowe said:
Hey sorry to return to this late but I am still pretty stuck on this. Dick, I don't think I know how to prove that

Better late than never. Try using (|a|-|b|)^2>=0.
 

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