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Test the series for convergence or divergence.

$ \displaystyle \sum_{n = 1}^{\infty} ( - 1)^{n-1} \frac {n^4}{4^n} $

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Converges

Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 7

Strategy for Testing Series

Sequences

Series

Missouri State University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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.

01:33

Test the series for conver…

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05:43

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00:55

two things toe observe here is that we're alternating sign and that we have polynomial growth on top and exponential growth at the bottom. So the alternating signed tests. We need to have that. Our terms are eventually strictly decreasing in absolute value to zero, so they can't be oscillating and then approaching zero, they need to be strictly decreasing to zero in absolute value. But since exponential growth is eventually going to be faster than polynomial growth, we will have eventually that the absolute value of how the terms in consideration will be strictly decreased into zero, which is what we want. So let me just write this down. Okay? So this is one of the things that we wanted to happen. We wanted for this limit, be zero. Okay, so that's important. And as we mentioned, it's also important that, you know, this is these are all positive. Thanks. Right, because we're just looking at the absolute value of this. So we want the absolute value to be decreasing zero like we're accomplishing here. So if we get this to happen and we're alternating signs than the alternating signed test gives us a convergence, Okay, so you should know that this is going to be a limit that goes to zero. Um, yeah, basically, because, yeah, this is polynomial, and this is exponential girls. But if you really want to be rigorous about it, you could always ah, apply Lopa towels rule, and then you do the derivative. Or if you do, Lopata was rule four times. Then the numerator is just goingto turn in tow some constant. And then we'll still have something that blows up to infinity and the denominator When you apply lope it'll four times and then you you'Ll see that this limit is indeed zero. But it should be intuitively obvious that exponential growth is faster enough that we're goingto accomplish this limit being zero.

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