Sequences, Series, and Limits Summary

Pre-Calculus · Axiom Academy

SUMMARY Sequences, Series, and Limits Three connected ideas that build the foundation of calculus — patterns, their sums, and the behavior they approach. A sequence is an ordered list of numbers: arithmetic sequences add a constant; geometric sequences multiply by a constant. A series is the sum of a sequence's terms. Finite series have closed-form sums; an infinite geometric series converges only when |r| < 1 . A limit describes the value a function (or a sequence of partial sums) approaches — even at a point it never reaches. These ideas are one chain: sequences define the terms, series add them up, and limits decide whether that sum settles down. Each term differs from the one before it by a constant common difference d . The terms grow linearly . Common difference: d = a_ n - a_ n-1 Pattern: add the same amount repeatedly Each term is the one before it times a constant common ratio r . The terms grow (or decay) exponentially . Common ratio: r = a_ n / a_ n-1 Pattern: multiply by the same factor repeatedly Add a fixed number of terms. Both kinds of sequence have an exact closed-form sum of the first n terms. The arithmetic sum is n times the average of the first and last terms. The geometric formula needs (otherwise just ). Add infinitely many terms. The sum converges to a finite value only when the terms shrink fast enough. A limit captures what f(x) approaches as x nears c — not necessarily the value at c .

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