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List the first five terms of the sequence.$ a_n = \frac {1}{n + 1}! $

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Calculus 2 / BC

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Series

Oregon State University

Idaho State University

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.

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List the first five terms …

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.List the first five terms…

Okay, so this should be written one over in plus one factorial, we're taking the factorial of n plus one, not taking the factorial of some fraction Mhm. Okay, so for an equals one, we have 1/2 factorial two factorial is just too. That's just one half. So you just need to remember that in factorial is equal to end times in minus one times in minus two all the way down two times one. Okay, so now we have for n equals two went over in plus one factorial. So that's 1/3 factorial. So three factorial is three times two, which is six for n equals three. We have one over four Factorial. Another way that we can do this is four Factorial is the same thing as four times three factorial. So four times six gives us 24. So that's just a shortcut you can use. So for in equals four, we have 1/5 factorial and again, five factorial is the same thing as five times four factorial. So five times 24 is on 20 for n equals five it's 1/6 factorial six factorial is six times five factorial. Hi. Effect. Oil is 1 20. So we get 1/1 70 here, so that gives us the first five terms in our sequence here.

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