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Determine whether the series is convergent or divergent. If it is convergent, find its sum.$ \displaystyle \sum_{n = 1}^{\infty} \arctan n $

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$$\frac{\pi}{2} \neq 0$$

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 2

Series

Sequences

Campbell University

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.

01:45

Determine whether the seri…

01:14

01:16

06:21

06:39

that's determine whether the Siri's convergence or diversions and then if it convergence will go ahead and find the sum, too. So some actual there? Possibly. So. Let's look at the limit and goes to infinity of armed men. The end. So before I write this answer, let me just refresh your memory. So to define Arc Tan, we just take the usual tan function. Believers just restricted between negative pyros to and power for two. So here's our usual tangent graph. So this graph, this will have an inverse because it passes the horizontal light test. And so, to graph the inverse recall that you just swap the X and Y values so these Assam talks will become horizontal ass from tops. So there's negative high over too. And then up here we have pie over, too. And now that we go ahead and let's guidance, for example, this point zero zero, if you switch those coordinates, you still get zero zero, so that will still be here. So go ahead and take each point on here and switched the X No, why in the following graph will be the this green one right here. So, by looking at the graph of arc tend, let's say ends. Of course, this girl, I'm drawing the continuous version. So let me just put an X here. We could see that the graft gets closer and closer to a pie or two. So that answers this question. Up here, our skin gets close to the survivors. Who and let's note that this is not equal to zero. And then we're basically finished here because we have what your book calls the diversions test. And this is if the limit as N goes to infinity of A M does not exist. No or if it does exist. But it's not equal to zero. That's what's happening in our case that exists. But it's not zero. Then we have that the Siri's diverges. Therefore, in our case, since we have that the limit is on zero, we have in conclusion and equals one to infinity. Our ten end is a diversion by the diversions test, and that's your final answer

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