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Determine whether the series is convergent or divergent. If it is convergent, find its sum.$ \displaystyle \sum_{n = 1}^{\infty} \ln \left( \frac {n^2 + 1}{2n^2 + 1} \right) $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 2
Series
Sequences
Oregon State University
Baylor University
Boston College
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.
02:19
Determine whether the seri…
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determined whether or not the Siri's convergence and if it is confusion will go ahead and find that some as well. So here this is definitely not geometric There it is possible to try the lot properties here to rewrite this. But before I do that, I will try the with the world calls the test for diversions. So let's go ahead and look at the limit of a N where Anne is just the things that are being added here is and goes to infinity So we just have the limit and goes to infinity natural log and square plus one, two, one squared plus one. So technically, I let me go ahead and yeah, you know what? Let me stick with me. Stick within the values of n So I'm gonna stick with this and then we can go ahead and we know how the log behaves. If you think of the log as being defined over the whole roll numbers, this is continuous function, so we could take the limit and put the limit on the inside. So this is just ln and then if you take the limit of this fraction on the inside, you can use, for example, open house rule. But if we take that limit, we give one half, and this is not equal to zero. Recall that Ellen of one equal zero and otherwise Allen of X is not equal to zero if X is not equal to one. And therefore what happens is when we've showed is that we've shown that limit of Ellen these air two conditions that need you need to show divergence. We showed that it exist. And then, too, we showed That's a limit and squared. Plus one No. Two, one squared plus one is not equal to zero. Therefore, this Siri's diverges by the and then I'LL just by the test for divergence. That's the reason for which the syriza emerges, and that's our final answer.
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