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Determine whether the sequence converges or diverges. If it converges, find the limit.$ \left \{ \frac {1}{1}, \frac {1}{3}, \frac {1}{2}, \frac {1}{4}, \frac {1}{3}, \frac {1}{5}, \frac {1}{4}, \frac {1}{6}, . . . \right \} $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 1
Sequences
Series
Harvey Mudd College
Baylor University
University of Michigan - Ann Arbor
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.
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.
03:20
$23-56$ Determine whether …
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Find the general term of t…
01:47
this problem, One of the hardest parts might just be figuring out exactly what the pattern is here. So for odd in are they in terms? Car one over one. Went over to one over three, one over four. And I'm sure you can see the pattern f and for even in Are they in terms? R one over three, one over four, one over five, one over six. And I'm sure you can see the pattern there as well. Okay, Both of these sequences converge to zero. Right. This sequence clearly goes to zero. This sequence clearly goes to zero. Therefore, the sequence sequence made up of just the's A in terms converges to zero. Right, because this is just a combination of both of these sequences here. So both those sequences converged to zero. Then the larger sequence must also converged to zero
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