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Use the Integral Test to determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} n^{-0.3} $

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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

Sawyer S.

November 17, 2021

The integral of x^-.03 is (x^.07)/.07

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

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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we can see that this function is a positive, decreasing function so we can use the interval test to do so. We we're going to evaluate the in cruel from one to infinity of X to the negative point three power. And of course, to solve this, we need to use a limit. And while we may initially think that because we have a negative exponents that we will and it was a converted, integral will find out it is different, Uh, in this case because when the anti differentiate But we'll still have a positive exponents. Yes, we have tea to the point seven power minus one and that will ah t to the point seven will go off to infinity. So that limit itself. It's infinity. So we know that the Siri's is then going to be a diversion

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