Comparison Tests
Compare with known series.
Key idea
Compare with known series.
In depth
If 0 ≤ aₙ ≤ bₙ and Σbₙ converges, so does Σaₙ. In the limit comparison test, if aₙ/bₙ tends to a positive finite limit, both series behave the same way.
Worked example
Σ 1/(n² + 1) ≤ Σ 1/n², which converges, so it converges.
Real-world uses
- Showing that the error of a series approximation is small by comparing with a known series
Common mistake
Comparing in the wrong direction: a larger divergent series says nothing.
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 Comparison Tests?
Compare with known series. It is a topic at the expert level in Calculus.
Where is Comparison Tests used in real life?
Real-world examples: Showing that the error of a series approximation is small by comparing with a known series.
What should I learn before Comparison Tests?
Make sure you are comfortable with Integral Test and p-Series, the previous topic in Calculus.
What is a common mistake with Comparison Tests?
Comparing in the wrong direction: a larger divergent series says nothing.
How can I practise Comparison Tests?
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 Comparison Tests?
Move on to Ratio and Root 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.