Integral Test and p-Series
Σ 1/nᵖ converges iff p > 1.
Key idea
Σ 1/nᵖ converges iff p > 1.
In depth
If f is positive, continuous, and decreasing, Σf(n) and ∫f(x)dx converge or diverge together. Consequently Σ 1/nᵖ converges when p > 1 and diverges when p ≤ 1.
Worked example
Σ 1/n² converges (p = 2); Σ 1/√n diverges (p = 1/2).
Real-world uses
- Testing whether a sum of decreasing values such as 1/n² is finite in statistics
Common mistake
Applying the test to functions that are not decreasing.
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 Integral Test and p-Series?
Σ 1/nᵖ converges iff p > 1. It is a topic at the expert level in Calculus.
Where is Integral Test and p-Series used in real life?
Real-world examples: Testing whether a sum of decreasing values such as 1/n² is finite in statistics.
What should I learn before Integral Test and p-Series?
Make sure you are comfortable with Divergence Test, the previous topic in Calculus.
What is a common mistake with Integral Test and p-Series?
Applying the test to functions that are not decreasing.
How can I practise Integral Test and p-Series?
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 Integral Test and p-Series?
Move on to Comparison Tests 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.