Proving Convergence: Bounded Sequences and Absolute Convergence Test

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In summary, if a sequence (bn) is bounded and the sum |(an)| going from n= 1 to infinity converges, then the sum of the product of sequences (an)(bn) converges. This can be proven by the absolute convergence test, which states that if the sum of the absolute value of a sequence converges, then the sum of the sequence also converges. Additionally, a bounded sequence is not necessarily convergent. However, if a sequence bn is bounded and convergent, then the sum of the product of bn and an will also converge. This can be proven by setting up a comparison test.
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Bellarosa
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1. Prove that if a sequence (bn) is bounded and the sum |(an)| going from n= 1 to infinity converges, then the sum of the product of sequences (an)(bn) converges.



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3. Given that the sequence bn is bounded it is convergent, and by the absolute convergence test if the sum of the absolute value of the sequence (an) converges then so does the sum of the sequence of (an), therefore the sum of the product of the sequences an and bn converges.
 
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A bounded sequence is not necessarily convergent.
 
  • #3
Oh yes that's true. So can I say that the sum of the product of a bounded sequence times a convergent sequence will be convergent? I'm assuming that the bounded sequence bn times the convergent sequence an will make the sum of their product convergent
 
  • #4
Bellarosa said:
Oh yes that's true. So can I say that the sum of the product of a bounded sequence times a convergent sequence will be convergent? I'm assuming that the bounded sequence bn times the convergent sequence an will make the sum of their product convergent

Sure you can 'say that'. Anybody can 'say that'. Aren't you supposed to 'prove that'? I would think about setting up a comparison test...
 

What is the definition of a bounded sequence?

A bounded sequence is a sequence in which all of the terms are contained within a certain range, meaning that there is a specific upper and lower limit for the terms in the sequence.

How do you determine if a sequence is bounded?

A sequence is considered bounded if you can find a number M such that all of the terms in the sequence are between -M and M. This means that the sequence is neither infinitely large nor infinitely small.

What is the Absolute Convergence Test?

The Absolute Convergence Test is a method used to determine if a series converges absolutely, meaning that the series converges regardless of the order in which the terms are added. This test is based on the idea that if the absolute values of the terms in a series converge, then the series itself must also converge.

How do you use the Absolute Convergence Test to prove convergence?

To use the Absolute Convergence Test, you must first take the absolute value of each term in the series. Then, you can use other convergence tests, such as the Comparison Test or the Ratio Test, to determine if the series of absolute values converges. If the series of absolute values converges, then the original series also converges absolutely.

Can you use the Absolute Convergence Test for all series?

No, the Absolute Convergence Test can only be used for series in which all of the terms are positive. If a series has both positive and negative terms, then the test cannot be used. In these cases, other convergence tests, such as the Alternating Series Test or the Root Test, may be used instead.

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