Sequences and Series Summary

Calculus 2 · Axiom Academy

SUMMARY Sequences and Series Summary How sequences approach limits, series sum infinitely many terms, and power series turn functions into polynomials — all decided by convergence. Infinity through limits: a series makes sense of an infinite sum by watching the partial sums S_n as . Convergence requires testing: terms shrinking to 0 is necessary but not sufficient — a battery of tests reveals the true behavior. Functions as polynomials: a power series represents a function as an infinite polynomial, enabling computation and analysis otherwise impossible. Local vs. global: a Taylor series matches every derivative at its center, but may only converge on a limited interval. Core Concept Sequences & Limits A sequence is an ordered list — a function a_n of a positive integer n . It converges to L when its terms get arbitrarily close to L as . Monotone convergence: a bounded, monotonic sequence always converges — it can neither oscillate nor escape. Watch out for: not every sequence converges — some oscillate forever, others grow without bound. Core Concept Series & Partial Sums A series adds every term of a sequence. It converges exactly when its sequence of partial sums S_n converges to a finite limit. Divergence test: if , the series must diverge — but the converse is false. Watch out for: does not guarantee convergence (the harmonic series). Core Concept Geometric & Special Series

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