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Determine whether the series is convergent or divergent.$ \frac {1}{3} + \frac {1}{7} + \frac {1}{11} + \frac {1}{15} + \frac {1}{19} + \cdot \cdot \cdot $
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Calculus 2 / BC
Chapter 11
Infinite Sequences and Series
Section 3
The Integral Test and Estimates of Sums
Sequences
Series
Ghasan M.
December 15, 2022
I think it is 1/4x-1
Missouri State University
University of Nottingham
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.
05:31
Determine whether the seri…
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00:45
since this is a continuous positive, decreasing function on the interval from one to infinity, we will use the integral test realizing their Siri's is one over to end plus one. So we will evaluate the integral from one to infinity of one over two X plus one suspect x, which means we have to take a limit. Uh, t ghost Infinity from wonder t of that function. And to integrate this we'LL do a u substitution Ah, where we will just have you equal to two x plus one which makes our do equal to two d x or in other words, one half to you is equal to the ex. So when we make our substitution, get the limit as T goes to infinity groom from some some coordinates here Andi won over you do you times one half We'LL take the one half hour in the front one half Really? We take the one half into the front of the integral. However property of limits is you can take constant coefficients in front limit as well. So just do it here for simplicity And when we integrate, we end up with the natural log of you evaluated from whatever those end points are. When we replace our exes back for youse, they will get one half the limit is he goes to infinity. Ah, Ellen. Two tea plus one minus Ellen of one. No, this whole difference in so gloomy. Ah, in between. In between these two steps I replaced the you with two x plus one from before and then plugged in the tea in the one from before. And as t goes to infinity Ah, Ellen to t plus one goes to infinity. So this whole, uh hola, mate goes to infinity. So because one half limit as t to infinity of the line of to t plus one minus on the one which is really just hero, that goes to infinity. So the siri's this diversion.
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