Alternating Series Test: Understanding the First Condition

In summary, the alternating series test states that an alternating series will converge if two conditions are met: the terms must be always positive and monotonically decreasing, and the limit of the nth term must go to 0 as n goes to infinity. This first condition may seem restrictive, but it is necessary for the series to converge. An example of an alternating series that meets both conditions but does not converge due to lack of monotonicity is shown to be 1, -1, 1/2, -1/2^2, 1/3, -1/3^2, and so on. This is because the positive and negative terms do not cancel out enough for the series to converge.
  • #1
LumenPlacidum
40
0
The alternating series test contains two conditions for convergence.

The first condition is that the nth term (extracting the power of -1) is always positive and monotonically decreasing.

The second is that the limit of that nth term goes to 0 as n goes to infinity.


I've seen a proof for it, and I've even proved it myself (although some years ago). I don't really understand why the first condition is so restrictive. It seems like the always-positive part is all you need combined with the limit going to 0.

Can someone show me an example of an alternating series for which the terms (again, without the power of -1) are always positive and that have a limit of 0, but which is NOT monotonically decreasing and diverges because of it?
 
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  • #2
[tex]
1,\; -1,\; \frac{1}{2},\; -\frac{1}{2^2},\; \frac{1}{3},\; -\frac{1}{3^2},\; \ldots
[/tex]

The sum of the positive parts tends to push the series towards [itex]+\infty[/itex], and the sum of the negative parts tends to push the series towards [itex]-\pi^2/6[/itex], so the negative parts cannot cancel the positive parts sufficiently for the series to converge.
 
  • #3
Great, thanks so much!
 

Related to Alternating Series Test: Understanding the First Condition

1. What are the alternating series conditions?

The alternating series conditions are a set of rules that determine whether a series, which is a sum of infinitely many terms, converges or diverges. These conditions apply specifically to alternating series, which are series where the signs of the terms alternate between positive and negative.

2. What is the first alternating series condition?

The first alternating series condition states that the absolute value of the terms in the series must decrease as the index of the terms increases. In other words, the terms must approach 0 as the index increases.

3. What is the second alternating series condition?

The second alternating series condition states that the limit of the absolute value of the terms must be 0 as the index approaches infinity. This means that the terms must eventually become smaller than any positive number as the index increases.

4. How do the alternating series conditions determine convergence or divergence?

If both alternating series conditions are met, then the series is said to converge. This means that the sum of the terms in the series approaches a finite value as the number of terms increases. If one or both of the conditions are not met, then the series is said to diverge, meaning that the sum of the terms either grows infinitely large or oscillates between different values.

5. Why are the alternating series conditions important in mathematics?

The alternating series conditions are important because they allow us to determine whether a series converges or diverges, which is crucial in many mathematical applications. They also provide a way to test for convergence without having to calculate the sum of an infinite number of terms. Additionally, many important series, such as the Taylor series, can be expressed as alternating series, making these conditions necessary for their use in mathematical calculations.

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