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Determine whether the series is convergent or divergent. If it is convergent, find its sum.$ \frac {1}{3} + \frac {1}{6} + \frac {1}{9} + \frac {1}{12} + \frac {1}{15} + \cdot \cdot \cdot $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Harvey Mudd College

Baylor University

Boston College

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.

02:28

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.

02:19

Determine whether the seri…

01:40

05:31

02:59

04:25

02:18

let's determine whether the Siri's is conversion or diversion, and if it is, conversion will go ahead and find some as well. So this Siri's, we can go ahead. And if it did converse, we should just be able to pull out of one over three and rewrite. This is and so on. However, the Siri's and the inside of the parentheses. We can rewrite this as the sum and equals one to infinity of one over end. If you want, you could think of this. One is one over one. But as mentioned in your textbook, this is a well known example. Call the harmonic series, and this one actually diverges. And if you take a diversion, Siri's and if you multiply it by one over three, it'LL still that verge. Though this whole thing here diverges, and therefore this sum that was given to us will also die average. So that version, and that's our final answer

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