/*! 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 64 Evaluate each finite series. $... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 64

Evaluate each finite series. $$\sum_{k=0}^{4} \frac{(-1)^{k} x^{k}}{k !}$$

Short Answer

Expert verified
The series evaluates to: \( 1 - x + \frac{x^2}{2} - \frac{x^3}{6} + \frac{x^4}{24} \).

Step by step solution

01

Understand the Series

The given series is \( \sum_{k=0}^{4} \frac{(-1)^{k} x^{k}}{k!} \). This means we will evaluate the sum for \( k = 0, 1, 2, 3, 4 \) by substituting each \( k \) into the expression \( \frac{(-1)^{k} x^{k}}{k!} \).
02

Calculate Each Term of the Series

Calculate for each \( k \):- For \( k = 0 \), \( \frac{(-1)^{0} x^{0}}{0!} = 1 \).- For \( k = 1 \), \( \frac{(-1)^{1} x^{1}}{1!} = -x \).- For \( k = 2 \), \( \frac{(-1)^{2} x^{2}}{2!} = \frac{x^2}{2} \).- For \( k = 3 \), \( \frac{(-1)^{3} x^{3}}{3!} = -\frac{x^3}{6} \).- For \( k = 4 \), \( \frac{(-1)^{4} x^{4}}{4!} = \frac{x^4}{24} \).
03

Sum of the Series

Add up all the evaluated terms: \[1 - x + \frac{x^2}{2} - \frac{x^3}{6} + \frac{x^4}{24}\] This sum represents the value of the series.
04

Final Result

The expression \( 1 - x + \frac{x^2}{2} - \frac{x^3}{6} + \frac{x^4}{24} \) is the final evaluated finite series. Substitute any specific value of \( x \) to get a numeric result or leave it as is for the general expression.

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Key Concepts

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

Understanding Factorials
Factorials are a fundamental concept in mathematics, especially when dealing with series and permutations. The factorial of a non-negative integer \( n \), denoted as \( n! \), is the product of all positive integers less than or equal to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).
  • The factorial operation grows very quickly as \( n \) increases. This rapid growth is often utilized in computation and combinatorics.
  • The factorial of zero is unique: \( 0! = 1 \). This is by definition and helps maintain consistency in mathematical formulas, especially evident in series expansions.
Factorials become particularly handy in series where terms involve shifts like powers or coefficients, helping to weigh terms differently.
Exploring Alternating Series
An alternating series is a series where the terms alternate in sign, switching back and forth. In the given exercise, \( (-1)^k \) causes the flip in signs, which is a key component of alternating series. Some characteristics of alternating series include:
  • The terms alternate between positive and negative.
  • These series can converge more rapidly due to the cancellation effects of adjacent terms.
In mathematical analysis, such series are critical because they often simplify complex expressions or models by canceling out significant portions of their terms, leading to easier convergence and simplicity in computation.
Characteristics of Polynomial Expressions
Polynomial expressions are mathematical phrases involving sums and products of variables and constants raised to non-negative integer powers. The expression from the exercise can be seen as a polynomial in \( x \):
  • Each term of the polynomial is in the form \( a_k x^k \), where \( a_k \) represents the coefficient of \( x^k \).
  • Polynomials are often evaluated at specific values of \( x \) to determine their usefulness in different contexts.
  • Polynomials are instrumental in approximating functions, simplifying patterns and solving equations in algebra.
The finite series given in the exercise is actually a truncated polynomial, showcasing its versatility in mathematics through simple operations.

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Most popular questions from this chapter

Use a graphing calculator to find \(\sum_{n=1}^{100}[-59+5(n-1)]\).

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