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Explain why the Integral Test can't be used to determine whether the series is convergent.$ \displaystyle \sum_{n = 1}^{\infty} \frac {\cos \pi n}{\sqrt n} $

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$a_{n}$ is not positive and decreasing

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 3

The Integral Test and Estimates of Sums

Sequences

Series

Campbell University

Harvey Mudd College

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.

01:39

Explain why the Integral T…

02:08

Explain why the integral t…

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01:21

the key requirement for the integral test is that function must satisfy three conditions. It must be continuous. And that is to say when our function is agonize that term of the what? When you can a continuous function, we need a decreasing function and we need and always positive function. Coastline of Pai n is well, the very first term is negative. Since co sign of pi times one is co sign a pie which is negative one. So this function is not always going to be positive and therefore it fails. Thie condition for General Test

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