Sum Math Problem: Find Condition for Finite Sum of Positive Real Numbers

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Discussion Overview

The discussion revolves around finding a condition under which the sum of a sequence of positive real numbers converges, given that the sum of the squares of the sequence is finite. Participants are exploring necessary and sufficient conditions related to convergence in the context of mathematical series.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Mathematical reasoning

Main Points Raised

  • One participant seeks to determine the condition that allows the convergence of the series \(\sum_{n=1}^{\infty} u_n\) given that \(\sum_{n=1}^{\infty} u_n^2\) is finite.
  • Another participant questions whether the original poster is looking for a necessary or sufficient condition for convergence.
  • A third participant suggests a theorem stating that the series \(\sum_{n=1}^{\infty} u_n\) converges if and only if \(\lim_{n \rightarrow \infty} (u_n \times n) = 0\), inviting others to prove this theorem.
  • The original poster expresses gratitude for the advice and indicates a willingness to revisit the problem.

Areas of Agreement / Disagreement

The discussion includes multiple viewpoints regarding the conditions for convergence, and no consensus has been reached on the specific condition required.

Contextual Notes

Participants have not clarified the definitions or assumptions underlying the conditions for convergence, and the discussion includes unresolved mathematical steps related to the proposed theorem.

mercedesbenz
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Let u_n be a sequence of positive real number.
If \sum_{n=1}^{\infty}u_n^{2} finite + (condition??) then \sum_{n=1}^{\infty}u_n finite.
I want to find the condition.Please help me.
 
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Are you looking for a necessary or a sufficient condition?
 
and real?
 
mercedesbenz said:
Let u_n be a sequence of positive real number.
If \sum_{n=1}^{\infty}u_n^{2} finite + (condition??) then \sum_{n=1}^{\infty}u_n finite.
I want to find the condition.Please help me.

IIRC, then there is a theorem like this:

Given the sequence of positive real number (un)

The series \sum_{n = 1} ^ {\infty} u_n converge, if and only if \lim_{n \rightarrow \infty}(u_n \times n ) = 0.

Let's see if you can prove this theorem. :)

Now, using the above theorem, can you try to work out the problem? :)
 
thank you so much for your advice,VietDao29.I will try to do it again.
 

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