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Determine whether the series converges or diverges.$ \displaystyle \sum_{n = 1}^{\infty} \frac {4^{n + 1}}{3^n - 2} $

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diverges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 4

The Comparison Tests

Sequences

Series

Oregon State University

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.

01:12

Determine whether the seri…

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determine whether this given series converges or diverges Here. I'll use the comparison test or direct comparison. The book might call it so this one does not require a limit. So here I'll just compare this some to another some. So first I'll use the fact that we have four n plus one over three to the N minus two. This is bigger than or equal to. Really. It's just strictly bigger than three to the end. And the reason for this? The reason the left side is bigger is because it's the nominator smaller. So therefore, yeah, the terms that were adding or smaller. This means that the entire sum is smaller. So I couldn't go ahead and replace the some with the smaller Some here because I'm using the inequality. And then I can rewrite this as a geometric series if I pull out the four. So here, just take this four out. So we have four to the end over three at the end, so I'll just write. That is four times for over three to the end power, and now we can see that this is geometric and we could even see that they are the value that we're continuing to multiply by each time we increase in. It's 4/3, however, are they satisfies this. The absolute value of our is bigger than one in any time. The absolute value of our is bigger than or equal to one. The series diverges. This is not our series. This is the geometric series. However, now we use the comparison test we have that our series is larger than the divergent series. So by the comparison test, our series also diverges bye, and I'll just abbreviate this by C t for a comparison test. Our series will also convert or also excuse me. Our series will also diverge because the series, the smaller series diverged and that's our final answer.

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