Question about the Integral Test

In summary, the conversation is about the difference between convergence and divergence in two examples shown on a math page. The speaker is confused about the concept and asks for an explanation. The expert summarizes that the difference lies in the value of x, with a series converging for x > 1 and diverging for x ≤ 1. The concept is demonstrated through the integral test. The speaker then realizes that this is a p-series representation, clarifying their confusion.
  • #1
yondy12
3
0
So this is my first post, I was wondering can you explain the first two examples of this page?

http://www.math.ubc.ca/~rathb/mar_6_p_4.jpg

What I don't understand is why, if there is a horizontal asymptote at p = 0.99 < 1 on first example, it diverges to infinity but in the second example, there is also a horizontal asymptote at p = 1.01 > 1 but it converges?

What's the difference and what concept am I not understanding here?

Thanks!
 
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  • #2
The convergence question is only distantly related to the asymptote.
A series with terms 1/nx diverges for x ≤ 1 and converges for x > 1.

The integral test shows this.
 
  • #3
mathman said:
The convergence question is only distantly related to the asymptote.
A series with terms 1/nx diverges for x ≤ 1 and converges for x > 1.

The integral test shows this.

Ah i realize this is simply a p-series representation. Thanks, that clears it up
 

1. What is the Integral Test?

The Integral Test is a method used to determine the convergence or divergence of an infinite series. It involves comparing the series to the integral of a related function.

2. How is the Integral Test used?

The Integral Test is used by evaluating the integral of a related function and then comparing it to the series in question. If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

3. What is the relationship between the Integral Test and the Divergence Test?

The Integral Test is a more powerful version of the Divergence Test. While the Divergence Test can only determine whether a series converges or diverges, the Integral Test can also provide an estimate for the sum of the series.

4. Are there any limitations to using the Integral Test?

Yes, the Integral Test can only be used on series with positive terms. Additionally, the function used in the integral must be continuous, positive, and decreasing on the interval of convergence.

5. Can the Integral Test be used for all infinite series?

No, the Integral Test can only be used for series that have positive terms and can be related to a continuous, positive, and decreasing function on the interval of convergence.

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