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Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} \frac {1 \cdot 3 \cdot 5 \cdot \cdot \cdot \cdot \cdot (2n - 1)}{2 \cdot 5 \cdot 8 \cdot \cdot \cdot \cdot \cdot (2n - 1)} $

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convergent

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Oregon State University

University of Michigan - Ann Arbor

Idaho State 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.

01:07

Test the series for conver…

02:19

01:08

0:00

01:33

01:39

03:05

Determine the convergence …

okay for this problem. I believe there's supposed to be a three here, not a two. So that might be one of the hardest problems for some people just recognizing this pattern down here. So we'll use the ratio test here. These terms here being r a in terms. So we're looking at limit as n goes to infinity of absolute value of and plus one over n. These terms of positive says the same thing as without the absolute value here. Okay, so this is limit as n goes to infinity one times, three times five times. Not that that time two and minus one And then we're looking at and plus one here. So we go one step further and similarly down below. We have two times, five times eight times that. That that And then we have times three and minus one and we're looking at a seven plus wan. So we go one step more than that. So all of this is just our ace of n plus one and we're dividing by a seven, which is the same thing as multiplying by the reciprocal of Ace have been so multiplying by two times five times a day, Not all the way up to three and minus one and then down below. We're having one times three times, five times that thought Times lips shed parentheses here, times two n minus one. Okay, so that one times three times, five times two in minus one. All of that is going to cancel out with all of this. Similarly, two times, five times, eight times. Not that thought Times three and minus one. All of that is going to cancel that with all of this. So now we have a limit as n goes to infinity. Two times in plus one minus one over three times in plus one minus one. And this limo is two thirds, which is less than one. So we do have convergence.

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