Problem 20 Use the binomial series to find ... [FREE SOLUTION]

Chapter 7: Problem 20

Use the binomial series to find the Maclaurin series for the function. $$ f(x)=\sqrt{1+x^{3}} $$

Short Answer

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The Maclaurin series for the function $f(x)=\sqrt{1+x^{3}}$ is $f(x) = 1 +\frac{1}{2}x^3 -\frac{1}{8}x^6 + ...$

Step by step solution

Identify the general form of the Binomial Theorem

The general form of the Binomial Theorem for any real number $a$ is: $(1+x)^a = 1 + ax + \frac{a(a-1)x^2}{2!} + \frac{a(a-1)(a-2)x^3}{3!} + ...$. This can be applied when $|x| < 1$, meaning that for this exercise, $|x^3| < 1$, which implies $|x| < 1$.

Apply the binomial theorem to the function

In our function, $f(x)=\sqrt{1+x^{3}}$, the term inside the square root follows the structure of the binomial theorem. Thus applying the theorem where $a = 1/2$ and replacing $x$ with $x^3$, we get: $f(x) = (1 + x^3)^{1/2} = 1 + \frac{1}{2}x^3 - \frac{1}{8}(x^3)^2 + ...$.

Write the Maclaurin series for the function

After applying the binomial theorem, the function $f(x)$ can now be expressed as a Maclaurin series: $f(x) = 1 +\frac{1}{2}x^3 -\frac{1}{8}x^6 + ...$. Note the Maclaurin series is simply a Taylor series expansion about zero.

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Key Concepts

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

Binomial series

The binomial series is a special case of the binomial theorem, which provides a way to expand expressions of the form $(1+x)^a$, where $a$ is any real number. In our exercise, we are interested in expanding $\sqrt{1+x^3}$, which is the same as $(1+x^3)^{1/2}$. This fits right into the form $(1+x)^a$, with $a = 1/2$ and $x$ replaced by $x^3$. When using the binomial series, it's key to ensure that $|x^3| < 1$, which leads to $|x| < 1$ for convergence. Some key aspects of the binomial series include:

This series is infinite, continuing indefinitely, with each term dependent on the powers of $x$.
Each new term in the series involves a higher power of $x$, multiplying by additional factors from $a$, decreasing by 1 each time, and divided by its factorial number.
This form makes it incredibly useful for approximations when $x$ is small.

The series can represent a wide variety of functions, particularly when they look like or can be rearranged into the form $(1+x)^a$.

Taylor series expansion

The Taylor series expansion allows us to express a function as an infinite sum of terms calculated from the values of its derivatives at a single point. Specifically, the Maclaurin series is a type of Taylor series expansion around the point zero. For the function $f(x) = \sqrt{1+x^3}$, we use the binomial series as a way to find the Taylor series expansion.Here's how the Taylor series expansion is structured:

Each term in the series involves the derivative of the function at a specific point (in Maclaurin, it's always 0), multiplied by powers of $x$.
The terms are adjusted by dividing by factorials to ensure convergence and accurate approximation.
The Maclaurin series is specifically about developing the Taylor series around $x=0$.

Using the binomial series gives us a shortcut in expanding $\sqrt{1+x^3}$ into a Maclaurin series without calculating individual derivatives.

Binomial theorem

The binomial theorem is a powerful tool in algebra, providing a formula for expanding expressions raised to any power. This theorem is written as:\[(1+x)^a = 1 + ax + \frac{a(a-1)x^2}{2!} + \frac{a(a-1)(a-2)x^3}{3!} + \ldots\] This is helpful in breaking down complex expressions into more manageable polynomial components.Key insights about the Binomial Theorem:

It shows the relationship between binomials raised to a power and polynomial expansions.
The coefficients of the expansion are derived from a combination of $a$, decreasing by one each term.

These are often visualized through patterns in Pascal's Triangle for integers, but it extends to real numbers as well in the binomial series.

The result is an infinite series when $a$ is not a whole number, which converges under certain conditions (like $|x| < 1$ in our solution).

By using the binomial theorem, we were able to quickly apply it to solve our exercise and derive the Maclaurin series, highlighting its versatility in mathematical analysis.

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91影视

Use the binomial series to find the Maclaurin series for the function. $$ f(x)=\sqrt{1+x^{3}} $$

Short Answer

Step by step solution

Identify the general form of the Binomial Theorem

Apply the binomial theorem to the function

Write the Maclaurin series for the function

Key Concepts

Binomial series

Taylor series expansion

Binomial theorem

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