Geometric series

A sum whose successive terms have a constant ratio. It can have finitely many terms or continue infinitely, with convergence depending on the ratio.

Common ratio: The fixed multiplier that takes each term of a geometric sequence to the next. It determines the terms and governs the series’ finite sum and infinite convergence.
Sequence: An ordered list of numbers or other mathematical objects indexed by position. The terms being added form an ordered sequence with a constant multiplicative step.
Compound interest: Interest calculated on both the original principal and previously accumulated interest. Repeated deposits or interest payments can produce sums whose terms grow by a fixed ratio.
Arithmetic series: A sum whose successive terms differ by a constant amount. Its terms change additively, unlike the constant multiplicative ratio here.
Finite geometric series: A geometric series containing a fixed, finite number of terms. Its sum follows from cancellation after multiplying the series by its common ratio.
Summation notation: A compact notation using the Greek letter sigma to indicate the sum of indexed terms. It expresses finite and infinite geometric sums by specifying their terms and index ranges.
Annuity: A sequence of equal payments made at regular intervals. Its present or future value is computed by summing payments scaled by a constant factor.
Harmonic series: The infinite sum of reciprocals of the positive integers. It diverges despite terms approaching zero, unlike convergent geometric series with ratio magnitude below one.
Infinite geometric series: A geometric series with infinitely many terms. Its partial sums approach a finite limit exactly when the common ratio has magnitude below one.
Exponentiation: The operation of raising a number to a power, representing repeated multiplication. The terms of a geometric series are commonly written as powers of its common ratio.

Linked from 37 pages