Infinite Series Summary

Real Analysis · Axiom Academy

Series convergence depends on how quickly terms approach zero. Absolute convergence ensures stability under rearrangement, while power series provide function representations. A series converges exactly when its sequence of partial sums approaches a finite limit — convergence requires the terms to shrink fast enough that the infinite accumulation stays bounded. Check the divergence test first, then pick a test by structure: geometric patterns → comparison, factorials → ratio, alternating signs → alternating series test. Rate of decay decides everything: p -series converge only when , because the terms must decay faster than . Absolute convergence is the gold standard — it lets you rearrange freely; conditional convergence is fragile (Riemann Rearrangement Theorem). Power series are "infinite polynomials" that represent functions inside their interval of convergence and can be differentiated and integrated term-by-term. Core Concept Series Convergence Fundamentals Definition: the series converges if the sequence of partial sums S_N approaches a finite limit. Necessary condition: if converges, then . The converse is false (harmonic series). Geometric series: converges to if , diverges otherwise. Key insight: convergence requires terms to shrink rapidly enough that the infinite accumulation remains bounded. Core Concept Essential Convergence Tests Divergence test: if , the series diverges. Quick first check. Comparison test: if and converges, then converges.

This is the written version of the interactive lesson above. See the full Real Analysis course.