Proving Convergence of Sum/Product of Non-Convergent Sequences

In summary, the conversation discusses how to prove that the sum or product of two non-convergent sequences can still be convergent. The speaker mentions trying various possibilities, such as a series times its reciprocal, adding a sequence to its negative sequence, and adding a function that goes to infinity to a function that goes to negative infinity. However, none of these attempts were successful in finding a suitable example. The conversation ends with a request for help in finding a suitable example.
  • #1
real analyst
10
0

Homework Statement



Prove by an example that the sum or product of two non convergent sequences can be convergent

Homework Equations



There are none, they can be any sequences I guess

The Attempt at a Solution



I've tried a lot of possibilities. My first guess would be a series times it reciprocal, but that just gives every term to be one, so, I don't know if that's really a good example. I also tried adding a sequence to i'ts negative sequence, but, that o course gives zero for every term. I don't think that's what he's looking for either. I also tried adding a function that goes to infinity to a function that goes to negative function, but I found that one function always outweighs the other, Any help would be appreciated.
 
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  • #2
well I'm sure you've shown that a_n = (-1)^n doesn't converge, now what's a_na_n?
 

FAQ: Proving Convergence of Sum/Product of Non-Convergent Sequences

How do you prove the convergence of a sum/product of non-convergent sequences?

To prove the convergence of a sum/product of non-convergent sequences, you must use the concept of limits. Specifically, you must show that the limit of the sum/product of the sequences is equal to the sum/product of the limits of the individual sequences. This can be done by using algebraic manipulations and the properties of limits.

Can you use the ratio test to prove convergence of a sum/product of non-convergent sequences?

No, the ratio test can only be used to determine the convergence of a single sequence. It cannot be applied to a sum/product of sequences.

Are there any other tests that can be used to prove convergence of a sum/product of non-convergent sequences?

Yes, there are other tests such as the comparison test and the root test that can be used to prove convergence of a sum/product of sequences. These tests can be applied to determine the convergence of the individual sequences and then the results can be used to determine the convergence of the sum/product.

Can a sum/product of non-convergent sequences ever converge?

Yes, it is possible for a sum/product of non-convergent sequences to converge. This can happen when the individual sequences have a specific pattern or relationship that allows for their sum/product to converge, even though the individual sequences do not converge.

Is it possible for a sum/product of non-convergent sequences to diverge?

Yes, it is possible for a sum/product of non-convergent sequences to diverge. This can happen when the individual sequences do not have a specific pattern or relationship that would allow for their sum/product to converge. In this case, the sum/product would also diverge.

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