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Problem

List the first five terms of the sequence. $ a_1…

01:12

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Problem 8 Easy Difficulty

List the first five terms of the sequence.
$ a_n = \frac {(-1)^nn}{n! + 1} $

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Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

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Lectures

01:59

Series - Intro

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

Sequences - Intro

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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Video Transcript

Kings were going to be working with the factorial. Similar. So let me just write down the relevant factorial values. One factorial is one two Factorial is too three Factorial is three times to which is six for Factorial is four times three times two So four times three factorial four times six is twenty four five. Fact world is five times for factorial one twenty. Okay, so these are the relevant factorial functions that we'll need for figuring out the terms in this sequence. For in equals one, we have minus one to the one. So some negative number negative one over one factorial plus one so minus one half That gives us the first term for an equals to We have minus one squared, so it'LL give us a positive number and then we have to in the numerator two divided by two factorial plus one So two over three, three nickels three We're back to having a negative number minus three over three factorial plus one two minus three over seven chronicles for backto having a positive number. We're just switching off signs here can. That's what this minus one to the end means over here. So four divided by four factorial plus one pour over twenty five. But I really should be putting commas toe separate values in our sequence here and then we just need one more term should be a negative negative five over five factorial plus one the negative five over one twenty one.

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