- Topics At a Glance
- Series: This is the Sum That Doesn't End
- Sigma Notation
- Alternating Series
- Convergence of Series
- Finally, Meaning...and Food
- Relationship Between Sequences
- Math-e-magics?
- When Limits of Summation Don't Matter
- Properties of Series
- Special Cases
- Arithmetic Series
- Geometric Series
- Finite Geometric Series
- Infinite Geometric Series
- Decimal Expansion
- Word Problems
- Visualization of Series
- When Limits of Summation Don't Matter
- Tests for Convergence
- The Divergence Test
- The Alternating Series Test
- The Ratio Test
- The Integral Test
- The Comparison Test
- Absolute Convergence vs. Conditional Convergence
- Summary of Tests
- Taylor and Maclaurin Series
- In the Real World
- I Like Abstract Stuff; Why Should I Care?
- How to Solve a Math Problem
Introduction to :
Since an uncontrolled grilled cheese spill is a hazardous materials catastrophe we'd like to avoid, we want a test that will tell us when not to open the Pandora's box.
We know that if a series converges, its terms must approach zero. Rephrasing this in the contrapositive, if the terms of a series don't approach zero, the series diverges. This statement lets us look at a series and, if the terms don't approach zero, conclude that the series diverges.
Be Careful: We can't use this statement to conclude that a series converges. We can only use it to evaluate if a series diverges. That's why we call it the Divergence Test. If the terms do approach zero, there's hope that the series might converge, but we would need to use other tools to really draw that conclusion.
Sample Problem
Does the series
converge or diverge?
Answer.
Look at the limit of the terms an = n as n goes to ∞. Since
the series diverges.
Practice:
Does the series
converge or diverge? |
Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.
The series 
converges.
,
converges" Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.
The series 
diverges.
diverges"Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.
The series 
might converge.
might converge"Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.
The series 
converges.
converges"Use only the Divergence Test to determine if the statement is true, false, or can't be decided yet.
The series 
diverges.
diverges"Use the divergence test to determine if the following series MUST diverge.
Use the divergence test to determine if the following series MUST diverge.
, diverges.Use the divergence test to determine if the following series MUST diverge.
. Since this limit is not equal to 0, the series diverges.Use the divergence test to determine if the following series MUST diverge.
. Since 
, this sequence might converge.Use the divergence test to determine if the following series MUST diverge.
