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**Series** are the sums of the terms of a sequence. Just like sequences, series can be finite or infinite. The main goal here is to determine whether a series converges (its sum approaches a finite value) or diverges (its sum does not approach a finite value). A series converges if the sum of its terms gets closer and closer to a finite value as we add more and more terms. If the sum grows indefinitely or oscillates, the series diverges. We will study several types of series, including arithmetic series (the sum of an arithmetic sequence) and geometric series (the sum of a geometric sequence). Geometric series are particularly important because they have a simple formula for convergence: If the absolute value of the common ratio is less than 1, the series converges. We will also introduce various tests for convergence, such as the comparison test, the ratio test, and the root test. These tests provide methods for determining whether more complex series converge or diverge. For example, the ratio test uses the ratio of consecutive terms to determine convergence. Series are used to represent functions, which is essential in physics, engineering, and computer science. Taylor and Maclaurin series, for example, represent functions as infinite sums of terms, providing powerful tools for approximating functions and solving complex equations.
