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Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = 3^n 7^{-n} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Oregon State University

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.

05:35

Determine whether the sequ…

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to figure out whether or not this sequence converges. There diverges. We need to look at the limit is and goes to infinity of a n And if the limit exists and is finite, then it converges. Otherwise it diverges. So we're looking at freed the end times seven to the minus and seven to the minus and is the same thing as one over seven to the end. Negative exponents make you take the reciprocal. So this is a limit. As in approaches, Infinity of three divided by seven to the end and three over seven is less than one in absolute value. That's what allows us to make this conclusion here. So it does converge and it converges to zero.

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