Series Convergence (Analysis)

Key idea

Partial sums converge to a limit.

In depth

A series converges if its partial sums converge. Tests (comparison, ratio, root, alternating, integral) are justified by the Cauchy criterion. Absolute convergence implies convergence.

Worked example

Σ 1/n² converges by comparison with Σ 1/(n(n − 1)), which telescopes.

Real-world uses

• Checking that infinite sums such as repeated payments or energy totals are finite

Common mistake

Using aₙ → 0 as a proof of convergence.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Analysis & Topology

• Axioms of the Real Numbers – ℝ is a complete ordered field.
• Completeness and Supremum – Every nonempty bounded-above set has a least upper bound.
• Archimedean Property – For any real x there is an integer n > x.
• Density of Rationals – Between any two reals there is a rational.
• Sequences and Convergence (ε–N) – aₙ → L if for every ε>0 eventually |aₙ − L| < ε.
• Limit Theorems for Sequences – Algebra of limits and squeeze theorem.