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Determine whether the series converges or diverges.$ \displaystyle \sum_{n = 1}^{\infty} \frac {6^n}{5^n - 1} $

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Diverges

00:28

Yiming Zhang

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Campbell University

Harvey Mudd College

University of Nottingham

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.

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Determine whether the seri…

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Tell whether the series co…

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Determine if the series be…

were given a series and were asked to determine whether this series converges or diverges series is the sum from n equals one to infinity of six to the end, over five to the end, minus one. Well, we know that this is a some with strictly positive terms and so we know that six to the end over five to the n minus one. This is also strictly greater than six to the end over five to the end for all natural numbers and therefore it follows the some itself is greater than the sum from n equals one to infinity of six to the end over five to the end. Now, this can also be written as 6/5 times the sum from and equals one to Infinity of six fits to the n minus first power. However, you want to look at it. Either way, we see that this is a geometric series with common ratio are equal to six bits. Of course, the absolute value of R is greater than one, so the geometric series diverges series on the right. Now, by the comparison test, a series greater than a divergent series also diverges. Therefore, the given series diverges as well, and for the comparison, we used a geometric series

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