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Geometric Series

Definition

A geometric series is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed non-zero number called the common ratio. The sum of all these terms forms an infinite geometric series.

Analogy

Think of a snowball rolling down a hill and getting bigger as it goes. Each time it rolls forward, it picks up more snow and becomes larger. In calculus, a geometric series works similarly - each term gets multiplied by a constant factor (like rolling forward), resulting in an ever-growing sum.

Related terms

Common Ratio: The common ratio refers to the fixed non-zero number that multiplies each term to get to the next one in a geometric sequence or series.

Convergent Geometric Series: A convergent geometric series is one where all its terms add up to a finite value when summed infinitely. This happens only when the absolute value of the common ratio is less than 1.

Divergent Geometric Series: A divergent geometric series occurs when adding up all its terms results in an infinite value. This happens when the absolute value of the common ratio is greater than or equal to 1.