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The terms of a series are defined recursively by the equations

$ a_1 = 2 $ $ a_{n+1} = \frac {5n + 1}{4n + 3} a_n $

Determine whether $ \sum a_n $ converges or diverges.

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The series diverges.

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 6

Absolute Convergence and the Ratio and Root Tests

Sequences

Series

Harvey Mudd College

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.

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02:41

given the sequence, and the first term is, too. And the following terms were given by the second equation here, and we'd like to know whether the sum of an conversions or diverges. So to do that, let's apply the ratio test, which requires that we look at the limit. So we're about the limit later, but the fraction is and plus one over and now, using our formula Pierre, we can write the numerator as such, and the denominator is just an now We could cancel those and terms, and we're just left with five plus one over foreign plus Tree I'LL drop the absolute values here because the remaining terms are all positive. Now we take the limit, and this let me goes to If you ply, let's say example low Patel's rule or some algebra you get five over for, which is bigger than one. So for this, we conclude that the Siri's A M diverges by the ratio test, and again, the reason we're saying diverges instead of converges is because the absolute value of and plus one over N and the limit is Shrink Lee larger than one

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