Alternating Series Test
Alternating decreasing terms tending to 0 converge.
Key idea
Alternating decreasing terms tending to 0 converge.
In depth
If terms alternate in sign, decrease in magnitude, and tend to 0, the series converges. The error from stopping early is at most the first omitted term.
Worked example
Σ (−1)ⁿ⁺¹/n converges (to ln 2). Partial sum error after 4 terms is below 1/5.
Real-world uses
- Estimating the error when summing alternating corrections in numerical methods
Common mistake
Forgetting to check that the terms decrease.
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 Alternating Series Test?
Alternating decreasing terms tending to 0 converge. It is a topic at the expert level in Calculus.
Where is Alternating Series Test used in real life?
Real-world examples: Estimating the error when summing alternating corrections in numerical methods.
What should I learn before Alternating Series Test?
Make sure you are comfortable with Ratio and Root Tests, the previous topic in Calculus.
What is a common mistake with Alternating Series Test?
Forgetting to check that the terms decrease.
How can I practise Alternating Series 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 Alternating Series Test?
Move on to Absolute vs Conditional Convergence 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.