Short answer
A sequence converges if its terms approach a finite limit. A series converges if its partial sums approach a finite limit. Tests like geometric, p-series, comparison, and ratio determine behavior.
- Sequence convergence and series convergence are related but different questions.
- A diverging nth term guarantees series divergence.
- Choose tests by term structure, not by memorized order.
Separate terms from partial sums
The sequence {an} tracks individual terms. The series sum an tracks accumulated partial sums Sn = a1 + ... + an.
A necessary condition for series convergence is an -> 0, but that alone is not sufficient.
Use standard convergence templates
Geometric series sum ar^(n-1) converges when |r| < 1. p-series sum 1/n^p converges when p > 1.
Comparison and ratio tests help with positive-term series that resemble known templates.
- If lim an != 0, the series diverges immediately.
- Know geometric finite sum and infinite sum formulas.
- State test conditions before applying them.
Determine convergence of a geometric series
Analyze sum from n=1 to infinity of (1/3)^n.
- Identify ratio r = 1/3 from term pattern.
- Since |r| < 1, the geometric series converges.
- Compute sum using first term a1 = 1/3 and formula a1/(1-r).
Result: Series converges to (1/3)/(2/3) = 1/2.
Common mistakes
- Assuming an -> 0 is enough to prove convergence.
- Using ratio test on forms where simpler tests are decisive.
- Mixing first term index when applying geometric sum formula.
- Not distinguishing conditional and absolute convergence contexts.
Try one
Does sum 1/n diverge or converge, and what named result supports it?
It diverges. This is the harmonic series, a standard divergence result.
Sources
- OpenStax Calculus Volume 1Open textbook covering limits, derivatives, applications, and introductory integration.
- College Board AP CalculusOfficial AP Calculus AB course overview and unit structure.