Calculus guide
Series and Sequences: Convergence Essentials
Distinguish sequences from series, recognize geometric and p-series, and choose a first convergence test.
10 min readIntermediateUpdated 2026-07-12
What you will learn
- Separate sequence and series questions
- Sum a convergent geometric series
- Use the nth-term and p-series tests
Concept 1
Sequences list terms; series add them
A sequence asks how aₙ behaves as n grows. A series asks whether the partial sums approach a finite value.
Concept 2
Geometric series depend on the common ratio
An infinite geometric series converges only when |r|<1. Its sum is the first term divided by 1−r.
Concept 3
The term test is necessary, not sufficient
If aₙ does not approach zero, the series diverges. If aₙ approaches zero, another test is still required. For p-series, convergence occurs exactly when p>1.
Worked example
Sum 1 + 1/2 + 1/4 + ⋯
- 1Identify a geometric series with a=1 and r=1/2.
- 2Check |r|<1, so the series converges.
- 3Use a/(1−r).
Answer
Common mistakes
- Assuming aₙ→0 guarantees convergence
- Using the finite geometric formula for an infinite sum
- Confusing the sequence aₙ with its partial sums
Check your understanding