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Determine whether the sequence converges or diverges. If it converges, find the limit.$ a_n = e^{-1/ \sqrt n} $

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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.

03:48

Determine whether the sequ…

01:54

01:35

once again we're looking at limit as n goes to infinity of Anne. If this limit exists and is finite in the sequence converges otherwise it's said to diverge. Since the exponential function is continuous, we're allowed to write this as e to the limit as n goes to infinity of minus one over squared of end. You're the continuous function. You can pull the limit inside of the function. So that's what we're doing here. And now this. This is something that we should know how to evaluate his in, goes to infinity, squared of and is going to go to infinity. So we're going to be looking at minus one over infinity, which is just like zero. So this turns into E to the zero r E to the minus zero, and that's just one. So this converges two, one

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