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Use the Root Test to determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 0}^{\infty} $ (arctan $ n)^n $

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Divergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Harvey Mudd College

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.

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Use the Root Test to deter…

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let's use the root test to determine whether the Siri's conversions air averages the roots has requires that we look at the limit and goes to infinity and through absolute value of an We're in our case, a N is given by this formula over here. So go ahead and use the following facts from algebra Regarding the end through, we could always rewrite this's X to the one over and Sol, go ahead and do that over here, not formula. So have ark tend of end to the end power and also we're dealing with absolute valley here. So let me just put the absolute value inside. And this is all being raised to the one of rent. And so we're dealing with the exponents in the following form. So we'LL use another fact about exponents when you raise an exponents. Ooh, another. You just multiply those two exponents. So here, when we multiply, be in the sea we just get one and then we just have the limit and approaches infinity absolute value arc tan. And if you could recall from the graph here when you take the limit of the Ark tan, it approaches one of its aspirin toads. So here's a rough sketch of the Ark tan graph. Lucy. So you can see from the graph of Arc Tan that is, the limit goes to infinity. The Y value approaches pi opportune. However, this number is larger than one, and so by the root tests we conclude that the Siri's diversions

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