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Definition ► A series is represented as the sum of a sequence of infinite terms. That is, a series is a list of numbers with addition operations between them. Ex. 1+1+1+1+……… ► In most cases of interest the terms of the sequence are produced according to a certain rule, such as by a formula, by an algorithm, by a sequence of measurements, or even by randomly generated numbers.
Infinite Series ► The sum of an infinite series a 0 + a 1 + a 2 + … + a n, where a 0 + a 1 + a 2 + … + a n are the terms of the sequence, is the limit of the sequence of partial sums S n = a 0 + a 1 + a 2 + … + a n, as n ∞, if and only if the limit exists. In other words it is the sum of S 1 = a 1 S 2 = a 1 + a 2 S 3 = a 1 + a 2 + a 3 …………… Where S 1, S 2, S 3, …, S n are the sum of the terms in the sequence.
Infinite Series (cont.) ► More formally, an infinite series is written as: where the elements in Sn are where the elements in Sn are real (or complex) numbers. real (or complex) numbers.
Convergence and Divergence ► If the sequence of partial sums reaches a definite value, the series is said to converge. ► On the other hand, if the sequence of partial sums does not converge to a limit (e.g., it oscillates or approaches +∞ or -∞), the series is said to diverge.
Convergence and Divergence (cont.) ► More formally, we say that the series converges to M, or that the sum is M, if the limit exists and is equal to M. If there is no such number, then the series is said to diverge.
Examples ► Convergent Series ► Divergent Series Geometric Series Harmonic Series
Series Handouts and Links ► Series Handout Series Handout Series Handout ► Sums and Series Handout Sums and Series Handout Sums and Series Handout ► Infinite Series Handout Infinite Series Handout Infinite Series Handout ► Solving Series Using Partial Fractions Handout Solving Series Using Partial Fractions Handout Solving Series Using Partial Fractions Handout ► Series Quiz Series Quiz Series Quiz