Proof of Convergence: a_n>=0 and summation a_n

In summary: Since 0< an2< |an|, since the series of |an| converges, by the comparision test, the series of an2 converges.In summary, if a_n>=0 and summation a_n converges, then summation (a_n)^2 also converges.
  • #1
fk378
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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.
 
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  • #2
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 [tex]a_{n}=\frac{1}{\sqrt{n}}[/tex]?
 
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  • #3
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)
 
  • #5
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 [itex]\sum a_n[/itex] converges then [itex]lim a_n= 0[/itex].
For sufficiently large n, an< 1 and so an2< |an|.
 

Related to Proof of Convergence: a_n>=0 and summation a_n

1. What is the definition of proof of convergence?

The proof of convergence is a mathematical technique used to show that a series, represented by summation a_n, converges to a finite value as n approaches infinity. This means that the sum of all the terms in the series approaches a specific value as the number of terms increases.

2. What does a_n>=0 mean in the context of proof of convergence?

This notation means that all terms in the series must be equal to or greater than zero. This is a necessary condition for proving convergence, as it ensures that the series does not have any negative terms that could cause the sum to diverge.

3. How do you prove convergence using the a_n>=0 condition?

To prove convergence, you must show that the series satisfies one of the convergence tests, such as the comparison test or the ratio test. These tests use the fact that a_n>=0 to determine whether the series converges or diverges.

4. Can a series with negative terms still converge?

No, a series with negative terms cannot converge. This is because the sum of an infinite number of negative terms will always be negative infinity, which is not a finite value.

5. How does the a_n>=0 condition impact the value of the sum of the series?

The a_n>=0 condition does not directly impact the value of the sum of the series. However, it ensures that the series converges to a finite value, which allows for the sum of the series to be calculated accurately.

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