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Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = \sqrt [n]{2^{1 + 3n}} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Missouri State University

Oregon State University

Harvey Mudd 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.

01:38

Determine whether the sequ…

01:35

02:49

So I think the easiest way to approach this problem is by rewriting this A in term. So this in that we see happening over there, That just means that all this stuff is being taken to the one over in power. So the one over n And then as long as you know how exponents work, then you know that this one over in is going to distribute to all of this stuff. So this turns into one times one over in plus three in times one over in, which is just three. So now if we do limit as n goes to infinity of a n mhm, we get limit as n goes to infinity of two to the one over N Plus three and to to the X is a continuous function, which means that you're allowed to pull the limit inside of the function. So this is the same thing as two to the power of limit as n goes to infinity of one over n plus three As n goes to infinity, one over N goes to zero and we get two to the three, which is eight. So our sequence converges and it converges to eight. Yeah,

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