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Determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} \frac {n}{n^4 + 1} $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 3
The Integral Test and Estimates of Sums
Sequences
Series
Oregon State University
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.
01:10
Determine whether the seri…
02:17
06:39
for this problem really is the comparison to and over end of the fourth plus one. It's always going to be less than n over just end of the fourth, because end of the fourth plus one is greater than end of the fourth. So dividing by a larger number, build a smaller number of all but end over end of the fourth reduces to one over and cute. We know that the series one over and Cubed converges as it is a P series. So, by comparison, test if our larger limit is this convergent series or sort of our larger terms come from this convergent series. In smaller terms of what we started with, we have convergent by comparison.
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