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Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} (\sqrt[n]{2} - 1)^n $

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converges.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Campbell University

University of Michigan - Ann Arbor

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.

0:00

Test the series for conver…

02:07

04:34

04:48

okay for this problem will let these guys b r a in terms and we'LL use the the root test. So for the root tests we look at Ltd's in goes to infinity of the in through of the absolute value of a m. Okay, so that's going to be lemma as n goes to infinity of and threw it of these guys. So the in through of this is just gonna get rid of the end as the exponents hair. So we'll just get the and through of two minus one. So remember that and threw two is just like two to the power of one over in. If you just rewrite it as n goes to infinity one over and goes to zero two, two zero is one. So this will be one minus one, which is zero. And the important part here is that this is something that's less than one So zero, certainly less than one. So we get convergence

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