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Use the Ratio Test to determine whether the series is convergent or divergent.

$ 1 - \frac {2!}{1 \cdot 3} + \frac {3!}{1 \cdot 3 \cdot 5} - \frac {4!}{1 \cdot 3 \cdot 5 \cdot 7} + \cdot \cdot \cdot $

$ ( - 1 )^{n-1} \frac {n!}{1 \cdot 3 \cdot 5 \cdot \space \cdot \cdot \cdot \space \cdot (2n - 1)} + \cdot \cdot \cdot $

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Absolutely Convergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Missouri State University

Harvey Mudd College

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.

04:49

Use the Ratio Test to dete…

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02:56

let's use the ratio test to determine whether the series converges. So, first of all, over here we see our end turn. This will be our A m. So we go to the formula for the ratio test Absolute value a n plus one over a m. So in our case, the only negatives that appear in the formula are out here in the front. We'll be taking absolute values so we could go ahead and it just ignore those. So the numerator. So here's my, uh I'm using this right now and then in the denominator, we have the product all the way up to this, and then we have the next term, which is when you replace and with n Plus One. And we still subtract one after that. So this is all my A M plus one. And now we'll go ahead and divide by a in here. So now let's go ahead and take that blue fraction and just flip it over and multiply it. So it's a product up there, not a minus sign and then And also let me simplify this last term over here. If you multiply that out, you'll get two n plus two and then minus one. And now, looking at this first fraction here, use the fact that n plus one factorial is the product of the first and plus one natural numbers. But if you combine the first end, you can write that as in factorial times and plus one. So let's go ahead and do that here. And the reason for doing that is we get nice cancellation here and also you can see and for the second fraction that we could cancel all the way up to two in minus one. And this is why it was helpful in the previous step that I that I wrote not just the final term of a M plus one, but I wrote the previous term because had I not written that it wouldn't appear over here. And it might be difficult to see how much you could actually get away with cancelling with these two terms here. Okay, so now we're just left with 1/2 n plus one. So finally limit and plus one, 21 plus one, use low petals rule if you'd like you get one half, which is less than one. So we conclude that the series converges Absolutely. But in this case, they just wanted convergent Oh yeah, mhm, Mhm. And there's our final answer.

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