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.List the first five terms of the sequence.$ a_n = \frac {n^2 - 1}{n^2 + 1} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Missouri State University

Campbell University

Harvey Mudd College

Baylor 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.

01:38

List the first five terms …

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Write the first five terms…

for in equals one, we have one squared, minus one over one squared, plus one. So we have zero over two, which is just zero. That's the first terminar sequence for in equals to we have two squared, which is four minus one over four. Plus one of three over five for n equals three, three squared, minus one over three squared, plus one. So that's nine minutes. Watches. Eight over nine plus watches. Ten. And this simplifies to four fifths. Yeah, you're in equals four. We have four squared, which is sixteen minus one, divided by sixteen plus one. So fifteen over seventeen and then for in equals five. This will give us the fifth term. In our sequence. We have five squared, minus one over five squared, plus one. So twenty four over twenty six, which simplifies to twelve over thirteen. Okay, so here is the first time three fifty is the next term. Four fifty is the next time. Fifteen over seventeen is the next term. Twelve over thirteen is the next term. And that gives us the first five times

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