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

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$\left\{\frac{2}{3}, \frac{4}{5}, \frac{8}{7}, \frac{16}{9}, \frac{32}{11}, \dots\right\}$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Missouri State University

Campbell 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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List the first five terms …

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.List the first five terms…

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03:48

Write out the first five t…

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01:49

Okay. Your terms, The sequence. We just need to plug in different values of N. For an equals one, we have to do the one over two times one plus one. So two thirds and equals to we have two squared over two times. Two plus one. So two over five. Sorry. Two squared over five. So for over five and equals three, you have the next power of to have top. Too cute. And then we'd have seven down here, you can pry. See the pattern. We have consecutive odd numbers happening down here. Three, five seven and powers of two up top. So Brandon equals four. We have sixteen. That's too to the fourth. Divided by two temps for plus one. So divided by nine in equals. Five, we have to the fifth thirty two, divided by two times five plus one divided by. So these are the first five term of our sequence

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