comparison test
Quick Summary
An infinite series is the sum of an endless sequence of numbers. These series are valuable tools in mathematics and various scientific disciplines.1
A partial sum is calculated by adding only the first few terms of an infinite series. If these partial sums approach a specific number as more terms are included, the series is said to converge. This specific number is then considered the sum of the series. If the partial sums do not approach a fixed number, the series diverges.12
For example, the series 1/2 + 1/4 + 1/8 + ... converges to 1. In contrast, the series 1 + 1 + 1 + ... diverges because its partial sums grow infinitely large.21
Tests exist to determine if a series converges or diverges, though not all series can be definitively classified. One such method is the comparison test, which is used when all terms in a series are positive. If a series' terms are less than or equal to the terms of a known convergent series, then the original series also converges.1
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