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Determine whether the series is convergent or divergent.$ \displaystyle \sum_{n = 1}^{\infty} \frac {1}{n^{\sqrt 2}} $

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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

Baylor University

University of Nottingham

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.

01:02

Determine whether the seri…

01:04

01:09

01:03

02:02

01:28

06:39

to determine whether the Siri's is conversion, not virgin. We can use the P Siri's test, which states that for a Siri's from an equals one to infinity of the form one over end the P power in the serious converges if P is greater than one. So we noticed in this case we have won over and to the square two. So you know, he is the square root of two. And we know that the square root of two has to be greater than the squared of one. And we know that squared of one is one. So in this case, we have p greater than one, meaning the serious his conversion.

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