Divergence Test
If aₙ does not tend to 0 the series diverges.
Key idea
If aₙ does not tend to 0 the series diverges.
In depth
If aₙ does not tend to 0, then Σaₙ diverges. The test can only prove divergence; if aₙ → 0 it says nothing about convergence.
Worked example
Σ n/(n + 1): terms tend to 1, so it diverges.
Real-world uses
- Quickly ruling out series that cannot converge when checking a model
Common mistake
Using it to claim convergence.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Divergence Test?
If aₙ does not tend to 0 the series diverges. It is a topic at the expert level in Calculus.
Where is Divergence Test used in real life?
Real-world examples: Quickly ruling out series that cannot converge when checking a model.
What should I learn before Divergence Test?
Make sure you are comfortable with Geometric Series Convergence, the previous topic in Calculus.
What is a common mistake with Divergence Test?
Using it to claim convergence.
How can I practise Divergence Test?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Divergence Test?
Move on to Integral Test and p-Series next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.