Proving Convergence of Sum(a_n)^2: Examples & Explanation

In summary, to prove that if a_n>=0 and sum(a_n) converges, then sum(a_n)^2 also converges, the only example necessary is 1/(n^p) where p is any exponent greater than 1. Other examples are not needed as using examples is not a valid method of proof. Instead, a comparison test between a_n and a_n^2 for n>=N can be used, where N is a positive integer. This is because if sum(a_n) converges, then there must be an N such that for n>=N, a_n<=1, and sum(a_n) for n>=N converges.
  • #1
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Homework Statement


Prove that if a_n>or equal to 0, and sum(a_n) converges, then sum(a_n)^2 also converges.


The Attempt at a Solution


The only example I can think of is 1/(n^p) where p is any exponent greater than 1. Are there any other examples that I can use or do I only need to use this example to prove it?
 
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  • #2
I don't think you can prove something by example.
 
  • #3
If sum(a_n) converges then there must be an N such that for n>=N, a_n<=1, right? sum(a_n) for n>=N converges. Do a comparison test between a_n and a_n^2 for n>=N.
 
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1. What is the definition of convergence in a series?

Convergence in a series refers to the property of a sequence of numbers or terms in a series to approach a specific limit or value as the number of terms increases.

2. How do you prove convergence of a series?

To prove convergence of a series, one must show that the limit of the partial sums of the series exists and is finite. This can be done through various methods such as the comparison test, ratio test, root test, or integral test.

3. What is the purpose of proving convergence of a series?

The purpose of proving convergence of a series is to determine if the series will converge to a specific limit or value, which can be useful in many mathematical and scientific applications.

4. Can you provide an example of proving convergence of a series?

Sure, an example of proving convergence of a series is the harmonic series, where the sum of the reciprocals of natural numbers is shown to diverge using the comparison test.

5. How does proving convergence of a series relate to the convergence of its square?

Proving convergence of a series can often be applied to the convergence of its square, as the convergence of the square can provide additional information about the convergence of the original series. This is particularly useful in cases where the original series may not converge, but its square does.

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