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Use the Root Test to determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} \left ( \frac {n^2 + 1}{2n^2 + 1} \right)^n $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Campbell University

Oregon State University

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.

01:13

Use the Root Test to deter…

01:40

03:03

01:43

02:48

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Use the comparison test to…

Let's use the roots has to determine whether the Siri's conversions or diverges. So let's go ahead and call this term am The root test requires that we look at the limit and goes to infinity, and then we take the end through of a N and let's see what this value is. So go ahead and replace and with our formula. And then instead of writing the end through I'LL use what sometimes is more convenient, rational explanations. So this is our a N and then now we'LL instead of writing the radical using this idea over here, we'LL just write this as a one over and power and we know that from our laws of exponents, if you have a next opponent, be and you raise that a b a to the P power to another exponents, you can go out and just multiply those two exponents of a So, in our case, when we won't supply the exponents, they just cancel each other out and leave us with one and we're left over with and square plus one two n squared, plus one. Now, if that helps, you can go ahead and replace and with X and use local house rule here. In either case, you see that this limit is one half this number is less than one. So the Siri's converges, and that's by the root test, and that's our final answer.

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