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Use the Integral Test to determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} \frac {2}{5n - 1} $

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The given series diverges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 3

The Integral Test and Estimates of Sums

Sequences

Series

Missouri State University

University of Michigan - Ann Arbor

Idaho State 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.

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Use the integral test to d…

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Use the Integral Test to d…

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So we have here a positive continues decreasing function, which means we can use the integral test. So we'LL take the integral from one to infinity to over five X minus one t x And of course, we need to change that into a limit and we're gonna use you substitution to do this integral. So we'll use the substitution with u equals five x minus one, which would make do you five d x. So this is really equal to the limit The studio's turn affinity any girl from one to infinity Or really, because we're changing the bounds, we should say so. Kind of a to b of too over you times to you over five The do over five comes from dividing the five from r D X over to the d you side. Now we can integrate giving us two fifths Allen of the absolute value of you. Ah, for the bounds of our into girls and replacing our ex right peace back in for you. We have this for integral. Now when we actually plug our bounds of integration, would that be should be a t We find that we end up with I two fifths, Alan five T minus one minus two fifths. Fell into four. No. And as t goes to infinity, this whole thing goes to infinity, so we know the Siri's is diversion.

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