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Determine whether the series is convergent or divergent.$ \frac {1}{5} + \frac {1}{7} + \frac {1}{9} + \frac {1}{11} + \frac {1}{13} + \cdot \cdot \cdot $

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DivergesHint: Use Integral test

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 3

The Integral Test and Estimates of Sums

Sequences

Series

Missouri State University

Baylor University

University of Michigan - Ann Arbor

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.

02:59

Determine whether the seri…

01:15

02:19

01:35

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00:45

$9-26$ Determine whether t…

01:06

were given a series and were asked to determine whether this series is convergent or divergent. So the series is 1/5 plus 1/7 plus 1/9 plus 1 11 plus 1 13 and so on. When I say and so on, what does this mean? That is, how would you continue find the next term in this some? Well, we see that we'd simply find the next odd number after 13, just 15 and take the reciprocal. So, nurse, we can write this as the sum from and equals, Let's say, zero to infinity of one over five plus to end course. This could also be written as these some from n equals one to infinity of one over two n Plus three doesn't really affect our method either way. Now, let's take ffx be the function 1/2 X plus three. Now we know that F of X is non negative on the interval from one to infinity. Indeed, it's non negative for all positive values of X. We also know that F is mon atomically decreasing on the interval from one to infinity. This is because it's an inverse function. In fact, it's decreasing for all X such that the denominator is non zero and therefore it follows that from the integral test follows that our series converges if and only if the integral from one to infinity of one over two. X plus three. The X converges. The question is, does this interval converge well and to go from one to infinity of 1/2 x plus three d x. This is the same as the limit as T approaches infinity of the integral from one duty of 1/2 x plus three d x, and this is equal to taking anti derivatives limit as T approaches infinity of one half times the natural log of two X plus three from X equals one t and plugging in. This is the limit as T approaches infinity of one half times the natural laws of two t plus three minus one half times the natural log of Just see, that's two times one is two plus three is five. Of course you know that limit This T purchase infinity of the natural log of two T plus three is again infinity. This is infinity minus one half natural log of five. Which of course is just infinity and therefore the integral diverges. And so it follows that these series also diverges by the integral test.

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