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

$ \frac {2}{3} + \frac {2 \cdot 5}{3 \cdot 5} + \frac {2 \cdot 5 \cdot 8}{3 \cdot 5 \cdot 7} + \frac {2 \cdot 5 \cdot 8 \cdot 11}{3 \cdot 5 \cdot 7 \cdot 9} + \cdot \cdot \cdot $

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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

Harvey Mudd College

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.

04:03

Use the Ratio Test to dete…

01:40

Determine whether the seri…

05:04

02:47

$7-24$ Use the Ratio Test …

05:31

02:59

to use the ratio test here. It would be helpful tohave the formula for the end term of the theories. Now we see here are first term a one a two and so on. And if we keep going on in this pattern you see in the numerator we start off with two and then we keep adding three. So it looks like we should have three and minus one. And you can check if an equals one. You too. If any calls to you get five and each time and increases the new writer and includes an additional factor of three increases by three. Now in the denominator, you see that we have three, five and so on and in the denominator or increasing by two. So this time we should have ah too, and and there instead. But when we plug in n equals one, we'd like to get three. So we should do two and plus one. So here's our formula for the INSERM. And so now we go to the ratio test This we look at the limit and goes to infinity and plus one over. And so now I could drop the absolute values because all of our numbers were positive. And now here, let's do a and plus one. So the numerator over here and then we would keep multiplying. And then we would include the next term. So that's the numerator for an plus one, Sol Do listen blue and the denominator and then we increase and buy one and I will divide buy n So we'Ll just use our formula here and now, as usual will go ahead and take that denominator flip it upside down and multiply it. So here, running out of a bit of room here. So let's see if we could squeeze this in and notice the very last term over here. If you simplify this, that's three n plus two in the denominator and then the very last term here, that's a two one plus three. No. Okay, now we're multiplying this by this thing after we flip it upside out and that will start canceling out as much as we can. So we look at the denominator over here and you see that you could cancel all the terms except the last one with this numerator over here. So that leaves us with a to N plus three. And the denominator. How about the numerator? It looks like we could cancel and then here. Sorry. That should have been a five. It looks like you could cancel everything up to three and minus one, and then you're left with to re n plus two. At this point, if it helps, you can use local tells rule toe, evaluate that limit. In either case, you get three over to which is bigger than one. So the Siri's diverges by the ratio test.

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