Proof of Convergence: a_n>=0 and summation a_n

  • Thread starter Thread starter fk378
  • Start date Start date
  • Tags Tags
    Convergence Proof
Click For Summary

Homework Help Overview

The discussion revolves around proving a statement related to the convergence of series, specifically that if \( a_n \geq 0 \) and the summation \( \sum a_n \) converges, then \( \sum (a_n)^2 \) also converges. Participants are exploring the implications of convergence and the behavior of sequences in this context.

Discussion Character

  • Conceptual clarification, Assumption checking, Mixed

Approaches and Questions Raised

  • The original poster attempts to establish a proof based on the limit of \( a_n \) and its implications for \( (a_n)^2 \). Some participants question the validity of this reasoning, particularly by introducing counterexamples like \( a_n = \frac{1}{\sqrt{n}} \). Others reference previous discussions and proofs related to the topic.

Discussion Status

The discussion is ongoing, with participants providing feedback on the original proof attempt. Some guidance has been offered regarding the limits and conditions for convergence, but there is no explicit consensus on a valid proof method yet.

Contextual Notes

Participants are navigating the complexities of convergence criteria and the implications of specific sequences. There is mention of previous discussions on the topic, indicating a broader context for the problem.

fk378
Messages
366
Reaction score
0

Homework Statement


Prove that if a_n>=0 and summation a_n converges, then summation (a_n)^2 also converges.

The Attempt at a Solution


(Note: When I say "lim" please assume the limit as n-->infinity). I just want it to be a little clearer to read)

If summation a_n converges, then lim(a_n)=0. If lim(a_n)=0, then lim(a_n)^2=0.
If summation (a_n)^2 diverges, then lim(a_n)^2 does not equal 0. But lim(a_n)^2=0, so summation (a_n)^2 must converge.

Can anyone let me know if this is a valid proof? I'm not sure how else to prove it otherwise...thank you.
 
Last edited:
Physics news on Phys.org
fk378 said:
If summation (a_n)^2 diverges, then lim (n-->inf) (a_n)^2 does not equal 0. But lim (n-->inf) (a_n)^2=0, so summation (a_n)^2 must converge.

I'm having some trouble following those lines. What about a_{n}=\frac{1}{\sqrt{n}}?
 
Last edited:
Oh, that does go against my proof. I don't know how else to prove it then. Any suggestions?

(I also edited a bit of my original post so that it would be a bit easier to read, hopefully)
 
fk378 said:
If summation (a_n)^2 diverges, then lim(a_n)^2 does not equal 0.
This is definitely NOT true!

However, it is true that if \sum a_n converges then lim a_n= 0.
For sufficiently large n, an< 1 and so an2< |an|.
 

Similar threads

Interval of convergence of power series from ratio test
  • · Replies 2 ·
Replies
2
Views
2K
On the ratio test for power series
  • · Replies 2 ·
Replies
2
Views
2K
How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K
If a series converges with decreasing terms, then na_n -> 0
  • · Replies 3 ·
Replies
3
Views
3K
Convergence of a recursive sequence
  • · Replies 3 ·
Replies
3
Views
2K
Is my proof that multiplication is well-defined for reals correct?
  • · Replies 5 ·
Replies
5
Views
2K
Showing that if lim a^2 = 0 implies lim a = 0
  • · Replies 5 ·
Replies
5
Views
2K
Convergent sequences might not have max or min
  • · Replies 7 ·
Replies
7
Views
2K
Do Convergent Ratios in Sequences Imply Equal Limits?
  • · Replies 7 ·
Replies
7
Views
2K
Convergence of a continuous function related to a monotonic sequence
  • · Replies 3 ·
Replies
3
Views
2K