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Test the series for convergence or divergence.$ \displaystyle \sum_{n = 1}^{\infty} (-1)^n \frac {n^n}{n!} $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 5

Alternating Series

Sequences

Series

Missouri State University

University of Nottingham

Idaho State University

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.

03:19

Test the series for conver…

04:27

02:01

00:40

0:00

02:13

01:24

01:06

00:55

just has this Siri's for conversions are diversions, although the Siri's is alternating, That doesn't necessarily mean we should go straight to the alternating theories test. Because here, if we call this being or in this case, let's not name this. Let's just look at this term here in the limit. I claimed that this limit is non zero. So first of all, if we just write this out as a fraction now, the numerator is just end end times. Whereas in the denominator, that's one times two times three all the way up. Ten. Now, if we look at each of these fractions, these fractions are all bigger than or equal to one. So that means that this entire product is bigger than or equal to just and over one. So this line may equals infinities. So, in other words, this lemon is bigger than or equal to the limit as n goes to infinity of end just by using this inequality over here and that's infinity. So what that tells me is that as and it's large a M, which is negative one to the end, this number will Oska lee between between large, larger and larger alternating values. So here I mean alternating inside. So this means that the limit of a N is not able to zero because the limit just it does not exist. So the Siri's diverges by the diversions test, and that's our final answer. So the Siri's I didn't write the word I shouldn't die emerges by. The divergence is, and that's the final answer.

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