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Use the Integral Test to determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} n^{-3} $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 3
The Integral Test and Estimates of Sums
Sequences
Series
Campbell University
Baylor University
University of Nottingham
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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Use the Integral Test to d…
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Use the integral test to d…
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to do this problem, We need to know that we have a continuous, positive, decreasing function and the function f f X equals X to the negative three. Or, in other words, one over X Cubed fits his criteria. It's graph looks something like this you're wondering. So now we can it say that the sum from one to infinity of end of the native three is less than any girlfriend. Want to infinity of one over X cubed dicks? So we shall now do is evaluate this integral and see if it's convergence. If it is convergent, then we have the seriousness conversion. So let's take a look. Since this is an improper integral, we're going to need to take a limit. And this is actually not too bad of an integral as funerals go, Ah, anti derivative, you know, in this year's nice power rule and we can we can already see where this is starting the head because X to the negative too is one over x squared. So this becomes negative. One half times one over t squared plus one half and we're taking the limit. Of course, as he goes to infinity and Steve goes to infinity. This term here goes to zero. So we're left with just one half. So the girl is convergent. And since the Siri's is less than the inner girl, we know Siri's itself is conversion.
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