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List the first five terms of the sequence.$ a_1 = 2, a_{n+1} = \frac {a_n}{1 + a_n} $

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$\left\{2, \frac{2}{3}, \frac{2}{5}, \frac{2}{7}, \frac{2}{5}, \dots,\right\}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Campbell 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.

00:50

List the first five terms …

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first term is given to us. They want us to. A two is a one over one plus one. So to over three Hey, through is a two over one, plus a two two thirds divided by one plus two thirds. And to simplify this, we can multiply the top on the bottom by three to get to over three plus two. Two over five four. It's a three over one, plus a three god, two over five, delighted by one plus two over five. And we can use the same type of trick to simplify. Here multiplied top and bottom by five. We get to delighted by five plus two to over seven and then a five. A four over one plus a four. A four was to over seven, and we simplify this by multiplying top bottom by seven. So we get to over seven plus two. So two overnight meccas us the first five terms in our sequence.

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