Infinite series with all negative terms

Click For Summary
SUMMARY

The discussion centers on the convergence of infinite series with negative terms, specifically the power series ∑(-1)n-1(xn/n). It is established that this series converges for the interval -1 < x ≤ 1, despite all terms being negative when x = -0.45. This challenges the conventional notion that convergence tests require positive terms. The example of the series -e is provided to illustrate that convergence can occur even with all negative terms.

PREREQUISITES
  • Understanding of power series and their convergence criteria
  • Familiarity with the concept of alternating series
  • Knowledge of convergence tests such as the Ratio Test and the Root Test
  • Basic calculus, particularly limits and series
NEXT STEPS
  • Study the properties of alternating series and their convergence
  • Learn about the Ratio Test and how it applies to series with negative terms
  • Explore examples of power series and their convergence intervals
  • Investigate the implications of convergence in series with all negative terms
USEFUL FOR

Mathematicians, students studying calculus, and anyone interested in the properties of infinite series and convergence criteria.

Ibraheem
Messages
51
Reaction score
2
Hello,

I have been reviewing my textbook lately, and I came across a rather paradoxical statements. all of the convergence tests in my book state that the terms of the series has to be positive. However, when I solved this power series ∑(-1)n-1(xn/n), I found that it converges for -1<x≤1, but when I plugged -0.45 to test if the infinite series will converge, I got all the terms negative. So my question is can a series converge if all its terms are negative?
 
Physics news on Phys.org
Yes. E.g. ##-e##.
 

Similar threads

Undergrad Infinite Series of Infinite Series
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Infinite Series - Divergence of 1/n question
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad Understanding the nth-term test for Divergence
  • · Replies 17 ·
Replies
17
Views
6K
High School Difficulty with understanding whether 1/n converges or diverges
  • · Replies 11 ·
Replies
11
Views
6K
Undergrad Convergence of an infinite series
  • · Replies 15 ·
Replies
15
Views
3K
Graduate Solving a power series
  • · Replies 3 ·
Replies
3
Views
4K
Undergrad Infinite series calculation of a potential
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad How does the ratio test fail and the root test succeed here?
  • · Replies 6 ·
Replies
6
Views
4K
Undergrad Method for finding a complex series?
  • · Replies 10 ·
Replies
10
Views
3K
High School Arithmetic mean of an infinite number of points?
  • · Replies 20 ·
Replies
20
Views
3K