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Determine whether the series converges or diverges.$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{\sqrt {n^2 + 1}} $

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Diverges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

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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.

01:07

Determine whether the seri…

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00:44

03:56

01:35

03:44

Determine if the series co…

let's determine whether the Siri's converges are diverges. Well, here we can go ahead and maybe compare this with another Siri's. So here, let's call this a M. And then let me compare this with BND equals one over the square root of and square, which happens to simplify it and just be won over it. Then let's use limit comparison test. So let's look at the limit as n goes to infinity and then we have a n over bien. So we take the limit as n goes to infinity. So if we do a n over bian, that should end up being and over the square root of n squared plus one. Now let me bring this limit over here. Since I'm running out of room, this is the limit and goes to infinity. Now what I'LL do here is I'LL go inside that radical Claudia and square So let's factor out and squared That's one plus one over and square. Then I pull the and square outside of the radical and then the square of and squared as we have seen already. That's just end. So then I could rewrite the denominator is n times one plus one over and square. Then we could cancel these ends. That's the whole point of factoring out the end of the radical. Now we just have the limit angles to infinity, one over a square room, one plus one over and square. The one over and squared will go to zero, and you just get one. So let me computer since says, if you take the limited and over being and you have a number, so this is positive. So we have a number one that satisfies this inequality. It's a positive number. It has to be bigger than zero, and it cannot be insanity, and that's what we have. So by living comparison test, the answer to our problem will be the same as the answer for the sum of Bien. Now we know the Siri's one over end this diverges. This is You can either use the pee test or just say it's a harmonic series. But if you use P equals one, then it's divergent by Peters, therefore, by Lim comparison. Since we got the answer and one in the limit, this tells us that our Siri's also diverges. So our Siri's diverges by Lim comparison says, and that's the final answer

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