Convergence

Definition

Convergence is the property of an infinite sequence, series, or integral where its terms or partial sums approach a specific, finite limit as the index approaches infinity.

$\lim_{n \to \infty} S_n = L$

• How to read: “The limit of the partial sum S sub n as n approaches infinity equals L.”
• Meaning / when to use: If this limit exists and is a finite number $L$, the infinite series converges to $L$.

Why It Matters

Convergence is the mathematical guarantee of stability. In fields like computer science and engineering, we rely on infinite series to approximate complex functions (like sine or exponential functions). Knowing a process converges ensures that an algorithm will settle on a stable, predictable answer rather than oscillating or crashing.

Core Concepts

• Sequence Convergence: A sequence $\{a_n\}$ converges if its individual terms settle on a single finite number.
• Series Convergence: A series $\sum a_n$ converges if its partial sums $S_n$ approach a finite value $L$.
• Necessary Condition: For a series $\sum a_n$ to converge, the individual terms must approach zero ($a_n \to 0$). Note that this is necessary but not sufficient (e.g., the Harmonic Series).
• Absolute vs. Conditional: A series is absolutely convergent if it converges even when all terms are positive. It is conditionally convergent if it only converges because of alternating signs.

Connected Concepts

• Divergence
• Alternating Series Test
• Convergence Rate
• Limit Laws Sequences