Convergence of an alternating series

In summary, the conversation discusses the convergence of a series given certain conditions on the sequence and its partial sums. If the limit of the partial sums approaches 0 for all values of m, then the series will converge. This is not mentioned in the provided proof and an additional condition of the sequence being alternating and decreasing in magnitude is required. The proof may be improved by using the sandwich theorem to show that the partial sum is always between two numbers that approach the same limit.
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Consider a sequence with the ##n^{th}## term ##u_n##. Let ##S_{2m}## be the sum of the ##2m## terms starting from ##u_N## for some ##N\geq1##.

If ##\lim_{N\rightarrow\infty}S_{2m}=0## for all ##m##, then the series converges. Why?

This is not explained in the following proof:
Screen Shot 2016-06-11 at 1.20.39 am.png

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Happiness said:
Consider a sequence with the ##n^{th}## term ##u_n##. Let ##S_{2m}## be the sum of the ##2m## terms starting from ##u_N## for some ##N\geq1##.

If ##\lim_{N\rightarrow\infty}S_{2m}=0## for all ##m##, then the series converges. Why?
This is not true, consider ##u_n=(-1)^n##. You also have to add that un are alternating and decreasing in magnitude.

The proof looks a bit sloppy, but you can use a similar approach to show that the partial sum is always between two numbers that approach the same limit. Sandwich theorem.
 
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1. What is the definition of convergence of an alternating series?

The convergence of an alternating series refers to whether or not the series approaches a finite sum as more terms are added. It is determined by the behavior of the series as the terms alternate between positive and negative values.

2. How is the convergence of an alternating series determined?

The convergence of an alternating series is determined by applying the alternating series test, which states that if the terms of the series decrease in absolute value and approach zero, then the series will converge. This test also requires that the terms alternate in sign and that the series is not strictly increasing or decreasing.

3. What is the difference between absolute and conditional convergence of an alternating series?

Absolute convergence refers to when an alternating series converges without regard to the signs of the terms, while conditional convergence refers to when the series only converges if the terms alternate in sign. A series that is absolutely convergent is also conditionally convergent, but the reverse is not always true.

4. Can an alternating series diverge?

Yes, an alternating series can diverge if it fails the alternating series test. This can happen if the terms do not decrease in absolute value or if the series is strictly increasing or decreasing.

5. How is the convergence of an alternating series related to other tests for series convergence?

The alternating series test is a specific case of the more general ratio test and comparison test for series convergence. However, it is particularly useful for alternating series because it only requires that the terms decrease in absolute value and approach zero, while other tests may have more stringent requirements.

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