/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 33 Find the first five terms of the... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 11: Problem 33

Find the first five terms of the infinite sequence whose nth term is given. $$c_{n}=\frac{(-1)^{n}}{(n+1) !}$$

Short Answer

Expert verified
The first five terms are -\frac{1}{2}, \frac{1}{6}, -\frac{1}{24}, \frac{1}{120}, and -\frac{1}{720}.

Step by step solution

01

Understand the Given Formula

The exercise gives the formula for the nth term of a sequence: \[ c_n = \frac{(-1)^n}{(n+1)!} \]Here, 'n' represents the term number in the sequence. '!' represents a factorial, which is the product of all positive integers up to that number.
02

Calculate the First Term (c_1)

Substitute n = 1 into the formula: \[ c_1 = \frac{(-1)^1}{(1+1)!} = \frac{-1}{2!} = \frac{-1}{2} Therefore, the first term (c_1) is -\frac{1}{2}. \]
03

Calculate the Second Term (c_2)

Substitute n = 2 into the formula: \[ c_2 = \frac{(-1)^2}{(2+1)!} = \frac{1}{3!} = \frac{1}{6} Therefore, the second term (c_2) is \frac{1}{6}. \]
04

Calculate the Third Term (c_3)

Substitute n = 3 into the formula: \[ c_3 = \frac{(-1)^3}{(3+1)!} = \frac{-1}{4!} = \frac{-1}{24} Therefore, the third term (c_3) is -\frac{1}{24}. \]
05

Calculate the Fourth Term (c_4)

Substitute n = 4 into the formula: \[ c_4 = \frac{(-1)^4}{(4+1)!} = \frac{1}{5!} = \frac{1}{120} Therefore, the fourth term (c_4) is \frac{1}{120}. \]
06

Calculate the Fifth Term (c_5)

Substitute n = 5 into the formula: \[ c_5 = \frac{(-1)^5}{(5+1)!} = \frac{-1}{6!} = \frac{-1}{720} Therefore, the fifth term (c_5) is -\frac{1}{720}. \]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

infinite sequence
An infinite sequence is a set of numbers arranged in a specific order, without any end. Unlike a finite sequence, which has a predefined number of terms, an infinite sequence goes on forever. Each number in the sequence is called a term. For instance, the sequence defined by the formula \( c_n = \frac{(-1)^n}{(n+1)!} \) generates an infinite number of terms as you keep substituting higher and higher values for 'n'.
  • The notation c\textsubscript{n} represents the nth term of the sequence.
  • The sequence continues indefinitely, meaning there's no last term.
Understanding infinite sequences is crucial because they can appear in various mathematical and real-world contexts, from series to scientific data analysis.
factorial
The factorial of a number, denoted as n!, is the product of all positive integers up to that number. For example, 4! (read as 'four factorial') is calculated as 4 \times 3 \times 2 \times 1 = 24.
  • 0! is defined as 1, because the product of an empty set is 1 by convention.
  • Factorials grow very quickly; for instance, 5! = 120 and 6! = 720.
Factorials are used in various mathematical contexts, including permutations, combinations, and the given sequence formula \( c_n = \frac{(-1)^n}{(n+1)!} \). In our sequence, the factorial is in the denominator, helping to scale down the value of each term significantly as n increases.
alternating series
An alternating series is a sequence where the terms alternate in sign. This means that the signs of the terms switch back and forth between positive and negative. The given sequence \( c_n = \frac{(-1)^n}{(n+1)!} \) is an example.
  • When n is odd, \( (-1)^n \) gives a negative term.
  • When n is even, \( (-1)^n \) gives a positive term.
Alternating series are important in mathematics because they often exhibit interesting convergence properties, meaning that the sum of their terms can approach a finite value even if the series itself has an infinite number of terms.
nth term
The nth term of a sequence is a formula that allows you to find the value of any term in the sequence based on its position number, n. In our exercise, the nth term is given by \( c_n = \frac{(-1)^n}{(n+1)!} \).
  • By substituting different values of n into the formula, you can calculate specific terms.
  • For example, when n=1, we have \( c_1 = \frac{(-1)^1}{(1+1)!} = \frac{-1}{2} \).
This concept is vital for sequences because it provides a way to generate terms systematically, ensuring you can always determine the next term or any term you need.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve each problem. What is the coefficient of \(a^{3} b^{7}\) in the expansion of \((a+b+2 c)^{10} ?\)

List all equally likely outcomes in the sample space for each of the following experiments. A coin and a six-sided die are tossed simultaneously (12 outcomes).

List all equally likely outcomes in the sample space for each of the following experiments. A pair of dice is rolled ( 36 outcomes).

Write the complete binomial expansion for each of the following powers of a binomial. $$\left(2 a+b^{2}\right)^{3}$$

Solve each probability problem. Tossing One Coin Twice If a single coin is tossed twice, then what is the probability of getting a. heads followed by tails? b. two tails in a row? c. heads on the second toss? d. exactly one head?
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.