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

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The given sequence converges to 0

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Baylor 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.

03:42

Determine whether the sequ…

02:46

00:30

first recall that the candidate function has a graph that looks like this. Oh, okay. The important part here is this segment here between minus pi over two and pie over to so notice that as we go up to infinity over here that the X value gets close to pie over too. Remember our can Our ten is the inverse of Qianjin between minus power two and power to Okay, So for the inverse functions the X and A Y coordinates air switched So here we go up to infinity X goes to poverty to so for our ten if we're looking at Ark Tan of infinity in the output would be pi over too can The important thing here is that this is just some finite number So when we're taking the limit as in goes to infinity of a end, that's limit as in goes to infinity of Ark tan we just use the same notation So this is just another notation for our tent art can of n divided by n We knew the trick where we dio limit on top over limit on bottom So we're allowed to do that as long as we don't get infinity over infinity or something divided by zero. So here, women on top, as we mentioned, that's going to be poverty Tuesday. Just some finite number limit on the bottom is going to be infinity. So this is some finite number divided by infinity, and that limit is zero. So we do converge. We converged to zero.

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