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Determine whether the series is convergent or divergent.$ 1 + \frac {1}{8} + \frac {1}{27} + \frac {1}{64} + \frac {1}{125} + \cdot \cdot \cdot $
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00:55
Carson Merrill
Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 3
The Integral Test and Estimates of Sums
Sequences
Series
Harvey Mudd College
Baylor University
University of Nottingham
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.
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01:35
in this problem, we have to determine if a Siri's is convergent or divergent. So the Siri's that were given isn't really in the form of a normal, normal Siri's with the Sigma. But we have to get it in that form to determine if it's convergent or not. So we're given this Siri's one plus 1/8 plus 1/27 plus 1/64 plus one over 125. Now, if we look at this, Siri's closely, we can see that we have a pattern here. So this is the same thing is saying that we have the some from n equals one to infinity of one over and cubed, so that looks like a form that we can test the convergence off. This is a part of me. This is a P Siri's, and in this case, P equals three and three is clearly greater than one. So that means that this Siri's is convergent by the P Siri's test. I hope that this problem help you understand a little bit more about Siri's, specifically how we can tell if a syriza's converging or not using the P Siri's test
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