Convergence and Comparison: Analyzing the Alternating Series Test

In summary, the conversation discusses a series and the use of the alternating series test (AST) to determine its convergence. It is mentioned that showing the function ln(n)/n is decreasing is necessary, and that this can be done by considering its derivative. The concept of absolute and conditional convergence is discussed, with the distinction being that absolute convergence is stronger and has more desirable properties. The conversation ends with a clarification that conditional convergence is not necessarily slower, but has different properties that make the distinction necessary for certain manipulations of the series.
  • #1
Bazzinga
45
0
So I have this series:

[tex]\sum^{infinity}_{n=3}(-1)^{n-1}\frac{ln(n)}{n}[/tex]

And I'm trying to use the AST to find out if it converges or not.
First of all, I'm stuck trying to show that ln(n)/n is decreasing...

But then after that. I'm assuming I can compare it with 1/n to show that it diverges absolutely, but converges conditionally (since the limit as n -> infinity of ln(n)/n is 0)

I was just wondering what converging absolutely and conditionally meant? We learned in class that absolute convergence implies convergence, does this mean that if its only conditionally convergent it doesn't converge?
 
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  • #2
Bazzinga said:
So I have this series:

[tex]\sum^{infinity}_{n=3}(-1)^{n-1}\frac{ln(n)}{n}[/tex]

And I'm trying to use the AST to find out if it converges or not.
First of all, I'm stuck trying to show that ln(n)/n is decreasing...

Consider the continuous extension of this function, i.e. consider the function

[tex]f:]0,+\infty[\rightarrow \mathbb{R}:x\rightarrow \frac{ln(x)}{x}[/tex]

it suffices to show that this function is decreasing (from a certain point on). To show this, it suffices to calculate the derivative of the function and seeing where it is negative.

But then after that. I'm assuming I can compare it with 1/n to show that it diverges absolutely, but converges conditionally (since the limit as n -> infinity of ln(n)/n is 0)

I was just wondering what converging absolutely and conditionally meant? We learned in class that absolute convergence implies convergence, does this mean that if its only conditionally convergent it doesn't converge?

Well, absolute convergence happens when the series of absolute values converge. Absolute convergence is stronger then convergence. However, there are series which converge and do not converge absolutely. This type of convergence is conditionally convergence.An example of this phenomenon is

[tex]\sum\frac{(-1}^n}{n}[/tex]

So series can do three kind of things: they can diverge, they can converge absolutely and they can converge conditionally.
 
  • #3
Ok, I understand that, but what's the difference between conditional convergence and absolute convergence? I always thought it was just 1s and 0s, it either converges or not, but conditional convergence is in between? Does is converge slower or something?
 
  • #4
No, there is nothing slower about conditional convergence (that I know of). Both conditional and absolute convergence is convergence. The only difference is that conditional convergent series do not converge absolute, while absolutely convergent sequences do.

The reason that mathematicians make the distinction is because there are a lot of nice properties of absolute convergent series that conditional convergent series do not have. For instance, in an absolute convergent series, you can put all the summands in an other place, and the series will still converge. Thus the convergence is commutative. However, a conditional convegent series is not commutative. There are some other distinctions, but you'll see them soon I guess...

If you're just interested in convergence of series, then there is no need for a distinction between conditional and absolute. However, if you want to do tricky things with the series, then such a distinction is necessairy!
 
  • #5
Ohh that's interesting! I guess I'll learn all that soon enough, thanks :)
 

1. What is the Alternating Series Test?

The Alternating Series Test is a mathematical test used to determine the convergence or divergence of an infinite series that alternates both positive and negative terms. It states that if the absolute value of the terms in the series decreases as n approaches infinity and the limit of the terms is 0, then the series is convergent.

2. How is the Alternating Series Test used to determine convergence?

To use the Alternating Series Test, you must first check that the terms in the series alternate between positive and negative. Then, take the absolute value of each term and check that the terms decrease in size as n approaches infinity. Finally, take the limit of the terms to see if it approaches 0. If all of these conditions are met, the series is convergent.

3. What is the difference between the Alternating Series Test and the Ratio Test?

The Alternating Series Test is a specific test used for alternating series, while the Ratio Test is a more general test that can be used for both alternating and non-alternating series. The Ratio Test also has more specific criteria for determining convergence, while the Alternating Series Test has fewer conditions to check.

4. Can the Alternating Series Test be used to determine divergence?

No, the Alternating Series Test can only be used to determine convergence. If the series does not meet the conditions for convergence, the test cannot be used to determine divergence. In this case, other tests such as the Divergence Test or the Integral Test must be used.

5. How is the Alternating Series Test applied in real-world situations?

The Alternating Series Test can be used to determine the convergence of infinite series that arise in various scientific fields, such as physics, engineering, and economics. For example, it can be used to determine the convergence of infinite series that model physical phenomena, such as the motion of a pendulum or the behavior of a fluid.

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