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Use the Ratio Test to determine whether the series is convergent or divergent.

$ \displaystyle \sum_{k = 1}^{\infty} \frac {1}{k!} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Harvey Mudd College

Baylor University

Idaho State University

Lectures

01:59

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

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In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

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Use the Ratio Test to dete…

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let's use the ratio test to determine whether the Siri's converges air averages. Now the ratio test requires that we look at this term a k one over k factorial. So we'd like to look at the limit as Kay goes to infinity. Absolute value, a k a. Plus one over a. K. So let's go ahead and write off this fraction in the numerator. That's one over Kaye plus one factorial have all in the denominator there. We just see a K. We already know what that is, so that goes down here in the denominator. And we couldn't drop the absolute value because the fractions are These are all positive numbers. So we have k factorial over Kaye plus one factorial. Let's go ahead and rewrite that denominator as que factorial times K plus one. Cross off those cave editorials and we just have limit one over Kaye plus one as Que Goes to infinity dis limit become zero. That's less than one. So we conclude at the original Siri's to sum from one to infinity converges by the ratio test, and that's our final answer

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