# Proof of Convergence: a_n>=0 and summation a_n

• fk378
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.
fk378

## 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:
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|.

## 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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