Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 9: Problem 22
Using a Binomial Series In Exercises \(21-26\) use the binomial series to find the Maclaurin series for the function. $$f(x)=\frac{1}{(1+x)^{4}}$$
Short Answer
Expert verified
The Maclaurin series for the function \(f(x)=\frac{1}{(1+x)^{4}}\) is \(f(x)= \sum_{k=0}^{\infty} \frac{(-1)^k 4.5.6...(3+k)}{k!} x^k \).
Step by step solution
01
Recall the Binomial Series
Invoke the Binomial Theorem or Series which is \( (1+x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k \) for \( |x| < 1 \). In this case, \(n\) is negative (-4).
02
Apply the Binomial Series to Function
Replace \(n\) with -4 in the Binomial Series definition, which will give us: \(f(x)= (1+x)^{-4} = \sum_{k=0}^{\infty} \binom{-4}{k} x^k\).
03
Simplify the binomial coefficient
To simplify the function further, we create general term based on the binomial coefficient pattern: \(f(x)= \sum_{k=0}^{\infty} \frac{(-4)(-5)...(-3-k)}{k!} x^k\).
04
Simplification of series terms
We can further simplify the series terms, leading to: \(f(x)= \sum_{k=0}^{\infty} \frac{(-4)(-5)...(-3-k)}{k!} x^k = \sum_{k=0}^{\infty} \frac{(-1)^k 4.5.6...(3+k)}{k!} x^k \).
05
Identify the Maclaurin Series
Maclaurin series is simply a Taylor series centered at 0. The general form of a Maclaurin series is \(f(x) = \sum_{n=0}^\infty \frac{f^n(0)}{n!} x^n \). Comparing this with the Binomial series identified earlier, the Maclaurin series corresponding to the given function is \(f(x)= \sum_{k=0}^{\infty} \frac{(-1)^k 4.5.6...(3+k)}{k!} x^k \).
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.
Binomial Series
The binomial series is a special type of power series that expands expressions of the form \((1 + x)^n\). This series is especially useful when \(n\) is not a positive integer, allowing it to handle cases with negative or fractional exponents. The general form of the binomial series is: \[(1 + x)^n = \sum_{k=0}^{\infty} \binom{n}{k} x^k\]where \(\binom{n}{k}\) represents the binomial coefficient.
- The series converges for \(|x| < 1\).
- It can be used to approximate functions in terms of polynomials.
- When \(n\) is a negative integer or a fraction, the expansion can be infinite.
Binomial Theorem
The binomial theorem provides a formula for expanding expressions raised to a power. It primarily applies to positive integer exponents, yet it serves as the foundation for understanding the binomial series. The theorem states that:\[(1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k\]where \(n\) is a non-negative integer. This differs from the binomial series by having a finite number of terms when \(n\) is a whole number.Key aspects include:
- \(\binom{n}{k}\) is the number of ways to choose \(k\) elements from \(n\) and is called a binomial coefficient.
- The coefficients correspond to those found in Pascal's Triangle, a useful tool for calculating smaller values.
- The binomial theorem simplifies calculations in cases where expansion is possible.
Taylor Series
A Taylor series represents a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Its versatility makes it a powerful tool in mathematical analysis. For a function \(f(x)\), the Taylor series centered around a point \(a\) is given by:\[f(x) = \sum_{n=0}^{\infty} \frac{f^n(a)}{n!} (x-a)^n\]where \(f^n(a)\) is the \(n\)-th derivative of \(f\) evaluated at \(a\).Important points include:
- The Taylor series allows for the approximation of functions through polynomial expressions.
- It converges to the function as more terms are added, typically within a specific range.
- It is crucial for understanding approximation techniques in engineering, physics, and other sciences.
Series Expansion
Series expansion involves expressing a mathematical function as an infinite sum of terms defined by a specific rule or pattern. This technique is integral in analyzing functions within a wide range of applications.
Some key characteristics include:
- Series expansions simplify complex functions into more manageable polynomial forms.
- They help in approximating functions that are difficult to compute directly.
- Applications range from solving differential equations to modeling physical systems.
