Convergence and the Alternating Series test

In summary, the conversation is discussing the possibility of a series converging without the constraint that a_n+1<=a_n. It is questioned whether a series can converge with only the requirement that a_n>0 and the limit as x approaches infinity=0, without being a decreasing monotonic series. If the answer is yes, the person is asked to provide three series that disprove the original conjecture, along with the mathematical proof of their divergence and an explanation of how they came up with the series. If the answer is no, the person is asked to provide a series that proves their conjecture, along with the mathematical proof of its convergence and an explanation of how they created the series. It is also mentioned that if this is
  • #1
deruschi12
1
0
Is it possible for a series to converge without the constraint that a_n+1< or equal to a_n? Can we have a convergent series with only the requirement a_n >0 and the limit as x approaches infinity = 0 (i.e. not a decreasing monotonic series)?

If yes include 3 series which disprove the original conjecture stated, the math that shows these series diverge, how you came up with your 3 series and what makes your series diverge

If no include the series to prove your conjecture, the math that shows your series converges, how you made your series and what makes your series converge
 
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  • #2
Is this a homework question? If so, it should be posted in the homework section, and you should show what you have done so far, and people will give you hints or help.
 

1. What is convergence in math?

Convergence in math refers to the behavior of a sequence or series as its terms approach a specific value or limit. It indicates whether the sequence or series will eventually stabilize or "converge" to a finite value, or if it will continue to fluctuate without ever reaching a definite value.

2. What is the Alternating Series test?

The Alternating Series test is a method used to determine the convergence of an alternating series, which is a series in which the terms alternate in sign. This test states that if the terms of an alternating series decrease in magnitude and approach zero, then the series will converge.

3. How do you apply the Alternating Series test?

To apply the Alternating Series test, you must first check if the terms of the series alternate in sign. Then, check if the absolute value of each term decreases as the series progresses. If both of these conditions are met, the series is convergent. However, if either of these conditions is not met, the series is divergent and the test cannot be used.

4. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series in which the terms decrease in magnitude and approach zero, regardless of their sign. This ensures that the series will converge to a specific value. On the other hand, conditional convergence refers to a series in which the terms alternate in sign, and the series will only converge if the terms decrease in magnitude. If the terms do not decrease in magnitude, the series will diverge.

5. Can the Alternating Series test be used to prove absolute convergence?

No, the Alternating Series test can only be used to prove conditional convergence. It cannot be used to prove absolute convergence because it only considers the behavior of the series when the terms alternate in sign. To prove absolute convergence, other tests such as the Ratio test or the Root test must be used.

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