Mr. Integral
Math notes/Calculus II/Sequences and Series

Sequences, Geometric and Telescoping Series — Cheat sheet

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Core rule

A series is the limit of its partial sums. ∑n=0∞arn=a/(1−r)\sum_{n=0}^\infty ar^n=a/(1-r) for ∣r∣<1|r|<1.

Watch for

Terms tending to zero do not prove convergence. Simplify telescoping finite sums before taking limits.