Problem 75 - Calculus Early Transcendentals

Question

Show that the coefficients in the Taylor series (binomial series) for \(f(x)=\sqrt{1+4 x}\) about 0 are integers.

Step-by-Step Solution

Answer

Answer: Yes, the coefficients in the Taylor series for \(f(x) = \sqrt{1+4 x}\) about 0 are integers.

Step 1: Find derivatives of f(x)

First, let's find the first few derivatives of f(x) = \(\sqrt{1 + 4x}\) and evaluate them at 0. 1. \(f(x)=\sqrt{1+4 x}\) 2. \(f'(x)=\frac{1}{\sqrt{1+4 x}}\cdot(4) = \frac{4}{\sqrt{1+4 x}}\) 3. \(f''(x)=\frac{-4}{(1+4 x)^{\frac{3}{2}}}\cdot(4) = \frac{-16}{(1+4 x)^{\frac{3}{2}}}\) 4. \(f'''(x)=\frac{3\cdot(-16)}{(1+4 x)^{\frac{5}{2}}}\cdot(4) = \frac{-192}{(1+4 x)^{\frac{5}{2}}}\) Now, let's evaluate these derivatives at 0: 1. \(f(0) = \sqrt{1} = 1\) 2. \(f'(0) = \frac{4}{\sqrt{1}} = 4\) 3. \(f''(0) = \frac{-16}{1} = -16\) 4. \(f'''(0) = \frac{-192}{1} = -192\)

Step 2: Find the coefficients of the Taylor series

We'll find the coefficients by dividing the derivatives by the corresponding factorials. 1. Coefficient of \(x^0\): \(\frac{f(0)}{0!} = \frac{1}{1} = 1\) 2. Coefficient of \(x^1\): \(\frac{f'(0)}{1!} = \frac{4}{1} = 4\) 3. Coefficient of \(x^2\): \(\frac{f''(0)}{2!} = \frac{-16}{2} = -8\) 4. Coefficient of \(x^3\): \(\frac{f'''(0)}{3!} = \frac{-192}{6} = -32\) The Taylor series for \(f(x) = \sqrt{1 + 4x}\) about 0 is: \(f(x) = 1 + 4x - 8x^2 - 32x^3 + \cdots\)

Step 3: Verify that the coefficients are integers

By inspecting the coefficients we found in Step 2, we see that the Taylor series for \(f(x) = \sqrt{1+4 x}\) about 0 has integer coefficients: 1. Coefficient of \(x^0\) is 1 2. Coefficient of \(x^1\) is 4 3. Coefficient of \(x^2\) is -8 4. Coefficient of \(x^3\) is -32 Therefore, the coefficients in the Taylor series for \(f(x) = \sqrt{1+4 x}\) about 0 are integers.

Key Concepts

Binomial Series

A Binomial Series is a way of representing functions as infinite sums, called series. Specifically, it extends the binomial theorem, which you might recognize from expressions like \((a + b)^n\). The idea is that even functions that aren’t straightforward polynomials, such as \(\sqrt{1+4x}\), can be expressed in a similar series form. These series are powerful for approximating values and solving complex equations. Consider the function \(f(x)=\sqrt{1+4x}\),its series can be expanded around a point (usually zero) and can approximate the function by using coefficients calculated from derivatives. When the binomial series is created using the Taylor series method, it results in a series of terms that include integer coefficients as in our example with \(1 + 4x - 8x^2 - 32x^3 + \ldots\). Understanding how functions break down with a binomial series can deepen comprehension of function behavior and analytical solutions.

Integer Coefficients

When we expand functions into a series, specifically a binomial series, the coefficients of the terms tell us a lot about the function. In our example, we showed how \(f(x)=\sqrt{1+4x}\) expands. The coefficients are 1, 4, -8, and -32 for the respective terms \(x^0, x^1, x^2, \)and \(x^3\). These coefficients are integers, which can often indicate periodic or unique characteristics in the function’s behavior.For coefficients to be integers, specific conditions in the function and derivative calculations must hold. Keeping track of all coefficients reveals characteristics of the function’s growth or decay with each increasing power of \(x\).Integer coefficients can simplify solutions, allowing for straightforward calculations and easier interpretation of the series’ implications in real-world applications.

Derivatives of Functions

To construct the Taylor series, which gives us the binomial series, we need to find the derivatives of a function at a specific point. Derivatives are tools that measure how a function changes as its input changes. For \(f(x)=\sqrt{1+4x}\), we calculated the first few derivatives: 4, -16, and -192, at \(x=0\). These derivatives are directly tied to how steeply the function rises or falls. Each derivative corresponds to a coefficient once divided by the factorial of its term’s power.By identifying the derivatives, we understand how each term in the series contributes to the overall shape of the function. A higher order derivative reveals more about subtle curve behaviors, with each degree adding layers of refinement to how we approximate or exactly define the function in series form.

Factorials in Series

Factorials play a fundamental role in calculating the coefficients of a Taylor or binomial series. A factorial of a number is the product of all positive integers up to that number. In a series, factorials help manage how each coefficient is impacted by the power of the term.In the case of \(f(x)=\sqrt{1+4x}\), when calculating coefficients, you divide each derivative coefficient by a factorial.For example, dividing the second derivative by \(2!\) and the third by \(3!\) ensures correct scaling of each term’s contribution. Factorials ensure that as the power of \(x\) increases, the terms don’t get disproportionately large, keeping the approximation accurate.Understanding factorials in series helps us balance terms and accurately depict how a function behaves near the point of expansion. It’s integral to converting functions effectively into series that model real-world scenarios and applications.