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Chapter 9: Problem 12

Find the binomial series for the functions. $$\left(1+x^{2}\right)^{3}$$

Short Answer

Expert verified
The binomial series for \((1 + x^2)^3\) is \(1 + 3x^2 + 3x^4 + x^6\).

Step by step solution

01

Define the General Binomial Theorem

The binomial theorem for expanding \((1 + x)^n\) for any integer \(n\), is given by:\[(1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k\]where \(\binom{n}{k}\) is the binomial coefficient, \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
02

Substitute into the Binomial Theorem

In this particular problem, we will substitute \(x^2\) for \(x\) and 3 for \(n\). This results in:\[(1 + x^2)^3 = \sum_{k=0}^{3} \binom{3}{k} (x^2)^k\]
03

Calculate Binomial Coefficients

Now calculate the binomial coefficients for \(n=3\):- \(\binom{3}{0} = 1\)- \(\binom{3}{1} = 3\)- \(\binom{3}{2} = 3\)- \(\binom{3}{3} = 1\)Make sure to use these coefficients to expand the expression.
04

Expand the Expression

Substitute the binomial coefficients and powers of \(x^2\) into the series:\[(1 + x^2)^3 = \binom{3}{0}(x^2)^0 + \binom{3}{1}(x^2)^1 + \binom{3}{2}(x^2)^2 + \binom{3}{3}(x^2)^3\]\[= 1(x^0) + 3(x^2) + 3(x^4) + 1(x^6)\]\[= 1 + 3x^2 + 3x^4 + x^6\]
05

Verify the Expansion

To verify, you can re-expand the expression step-by-step or use a calculator to ensure that multiplying out \((1+x^2)(1+x^2)(1+x^2)\) also results in \(1 + 3x^2 + 3x^4 + x^6\).

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Key Concepts

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

Binomial Theorem
The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions of the form \((1 + x)^n\), where \(n\) is any non-negative integer. At its core, the theorem depicts how each term in the expansion is derived.
The expression is represented in a compact form using a summation, as follows:
  • \((1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k\)
Here, \(\binom{n}{k}\) denotes the binomial coefficient and indicates the number of ways to choose \(k\) elements from \(n\) elements, which is crucial for determining the coefficients of each term. This theorem is essential for simplifying and solving polynomial expressions without having to manually multiply each term.
Binomial Coefficients
Binomial coefficients are a key component of the Binomial Theorem and represent the weights of each term in the polynomial expansion. They are denoted by \(\binom{n}{k}\) and are calculated using factorials:
  • \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\)
For example, in our exercise where \(n = 3\), the coefficients for each term are computed as follows:
  • \(\binom{3}{0} = 1\)
  • \(\binom{3}{1} = 3\)
  • \(\binom{3}{2} = 3\)
  • \(\binom{3}{3} = 1\)
These coefficients ensure that the polynomial is expanded correctly, preserving the balance of terms.By applying these coefficients, we systematically build each expanded term from the original binomial expression.
Polynomial Expansion
Polynomial expansion using the Binomial Theorem effectively breaks down a binomial raised to any power into a series of terms. This is especially helpful in simplifying complex expressions.
In our particular example, \((1 + x^2)^3\) is expanded through replacement of \(x\) with \(x^2\), while maintaining the same binomial structure:
  • First term: \(\binom{3}{0}(x^2)^0 = 1\)
  • Second term: \(\binom{3}{1}(x^2)^1 = 3x^2\)
  • Third term: \(\binom{3}{2}(x^2)^2 = 3x^4\)
  • Fourth term: \(\binom{3}{3}(x^2)^3 = x^6\)
Together, these terms form the complete polynomial:
\(1 + 3x^2 + 3x^4 + x^6\).
This method showcases the efficiency of the Binomial Theorem, simplifying what could otherwise be a laborious calculation.

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Most popular questions from this chapter

Which of the series, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{(-1)^{n}(3 n) !}{n !(n+1) !(n+2) !}$$

Use any method to determine whether the series converges or diverges. Give reasons for your answer. $$\sum_{n=1}^{\infty} \frac{(2 n+3)\left(2^{n}+3\right)}{3^{n}+2}$$

The Taylor polynomial of order 2 generated by a twice-differentiable function \(f(x)\) at \(x=a\) is called the quadratic approximation of \(f\) at \(x=a\). In Exercises \(45-50,\) find the (a) linearization (Taylor polynomial of order 1 ) and (b) quadratic approximation of \(f\) at \(x=0.\) $$f(x)=\cosh x$$

Two complex numbers \(a+i b\) and \(c+i d\) are equal if and only if \(a=c\) and \(b=d .\) Use this fact to evaluate $$\int e^{a x} \cos b x d x \quad \text { and } \int e^{a x} \sin b x d x$$ from $$\int e^{(a+i b) x} d x=\frac{a-i b}{a^{2}+b^{2}} e^{(a+i b) x}+C$$ where \(C=C_{1}+i C_{2}\) is a complex constant of integration.

Which of the series, and which diverge? Give reasons for your answers. $$\sum_{n=1}^{\infty} \frac{2^{n} n ! n !}{(2 n) !}$$
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