Question about the Integral Test

Click For Summary
SUMMARY

The discussion centers on the Integral Test for convergence of series, specifically p-series. It clarifies that a series with terms of the form 1/n^x diverges for x ≤ 1 and converges for x > 1. The confusion arises from the relationship between horizontal asymptotes and convergence, which is addressed by emphasizing that the asymptote does not directly determine convergence. The Integral Test is confirmed as the appropriate method for evaluating these series.

PREREQUISITES
  • Understanding of p-series and their convergence criteria
  • Familiarity with the Integral Test in calculus
  • Knowledge of horizontal asymptotes in mathematical functions
  • Basic concepts of series and sequences
NEXT STEPS
  • Study the Integral Test in detail, focusing on its application to various series
  • Explore p-series and their convergence properties
  • Learn about horizontal asymptotes and their implications in calculus
  • Practice problems involving convergence and divergence of series
USEFUL FOR

Students of calculus, educators teaching series convergence, and anyone seeking to deepen their understanding of the Integral Test and p-series.

yondy12
Messages
3
Reaction score
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!
 
Last edited by a moderator:
Physics news on Phys.org
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.
 
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
 

Similar threads

Undergrad Testing for Divergence using the Integral Test
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Understanding the nth-term test for Divergence
  • · Replies 17 ·
Replies
17
Views
6K
Undergrad Looking for help to understand the Integral Test
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad How does the ratio test fail and the root test succeed here?
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Infinite Series - Divergence of 1/n question
  • · Replies 5 ·
Replies
5
Views
3K
High School Integration by parts of inverse sine, a solved exercise, some doubts...
  • · Replies 4 ·
Replies
4
Views
3K
High School Question about the limits of integration in a change of variables
  • · Replies 2 ·
Replies
2
Views
3K
MHB Is the series $ \sum_{n=2}^{\infty} \frac{1}{n^2} $ converges?
  • · Replies 8 ·
Replies
8
Views
4K
Undergrad Express the limit as a definite integral
  • · Replies 16 ·
Replies
16
Views
4K
High School Understanding about Sequences and Series
  • · Replies 3 ·
Replies
3
Views
2K