Problem 68 - Calculus Early Transcendentals

Question

a. Use any analytical method to find the first four nonzero terms of the Taylor series centered at 0 for the following functions. In most cases you do not need to use the definition of the Taylor series coefficients. b. If possible, determine the radius of convergence of the series. $$f(x)=b^{x}, \text { for } b > 0, b \neq 1$$

Step-by-Step Solution

Answer

The first four nonzero terms of the Taylor series are: $$T_4(x) = 1 + (\ln{b})x + \frac{(\ln^2{b})x^2}{2!} + \frac{(\ln^3{b})x^3}{3!} + \frac{(\ln^4{b})x^4}{4!}$$ The radius of convergence for this Taylor series is infinite.

Step 1: Calculate the first four derivatives of f(x)

To compute the first four derivatives, we differentiate $$f(x) = b^x$$ four times with respect to x: $$f'(x) = \frac{d(b^x)}{dx} = b^x \ln{b}$$ $$f''(x) = \frac{d^2(b^x)}{d^2x} = b^x \ln^2{b}$$ $$f'''(x) = \frac{d^3(b^x)}{d^3x} = b^x \ln^3{b}$$ $$f^{(4)}(x) = \frac{d^4(b^x)}{d^4x} = b^x \ln^4{b}$$

Step 2: Evaluate the derivatives at the center point 0

Now we evaluate each derivative at the center point 0: $$f(0) = b^0 = 1$$ $$f'(0) = b^0 \ln{b} = \ln{b}$$ $$f''(0) = b^0 \ln^2{b} = \ln^2{b}$$ $$f'''(0) = b^0 \ln^3{b} = \ln^3{b}$$ $$f^{(4)}(0) = b^0 \ln^4{b} = \ln^4{b}$$

Step 3: Build the Taylor series¶

Using the derivatives evaluated at the center point, we can construct the Taylor series up to the first four nonzero terms: $$T_4(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \frac{f^{(4)}(0)x^4}{4!}$$ $$T_4(x) = 1 + (\ln{b})x + \frac{(\ln^2{b})x^2}{2!} + \frac{(\ln^3{b})x^3}{3!} + \frac{(\ln^4{b})x^4}{4!}$$

Step 4: Determine the radius of convergence¶

To find the radius of convergence, we can use the Ratio Test on the Taylor series: $$lim_{n\to\infty}\frac{(\frac{(\ln^{n+1}{b})x^{n+1}}{(n+1)!})}{(\frac{(\ln^{n}{b})x^{n}}{n!})}=lim_{n\to\infty}\frac{\ln{b}x}{n+1}$$ Since the limit is 0 for all x as $$n \to \infty$$, and $$\ln{b} \neq 0$$ (since $$b \neq 1$$), the Taylor series converges for all x (the radius of convergence is infinite). #Answer# The first four nonzero terms of the Taylor series centered at 0 for the function $$f(x) = b^x$$ are: $$T_4(x) = 1 + (\ln{b})x + \frac{(\ln^2{b})x^2}{2!} + \frac{(\ln^3{b})x^3}{3!} + \frac{(\ln^4{b})x^4}{4!}$$ And the radius of convergence for this Taylor series is infinite.

Key Concepts

Derivative

Understanding the concept of a derivative is crucial for working with Taylor series. A derivative represents the rate at which a function changes at any given point. It is essentially the slope of the tangent line to the function at that point. To find derivatives of a function like \( f(x) = b^x \), differentiate it with respect to \( x \). Each time you differentiate, you get a higher order derivative, which plays a key role in constructing Taylor series terms.

For instance:

• The first derivative, \( f'(x) \), indicates how \( f(x) \) changes linearly at each point.
• The second derivative, \( f''(x) \), gives information on the curvature of the function.
• Higher-order derivatives, such as \( f'''(x) \) and \( f^{(4)}(x) \), provide even more precise data on how the function behaves.

These derivatives are instrumental in determining each term's coefficient in the Taylor series, making them essential for approximating functions accurately.

Radius of Convergence

The radius of convergence is a critical concept when dealing with power series like the Taylor series. It tells us the values of \( x \) for which the series converges to the function it represents. In simple terms, it's like drawing a circle around the center point of a Taylor series (in this exercise, 0), within which the series reliably approximates the function.

For the function \( f(x) = b^x \), calculating the radius of convergence involves determining where the series remains valid and useful. Fortunately, for exponential-type functions like these, the radius turns out to be infinite. This means that no matter how far you move away from the center point 0, the series will converge to the actual function. Understanding this concept is important as it ensures the approximation is accurate within an expected range.

Ratio Test

The Ratio Test is a mathematical technique used to determine the convergence of a series. It's particularly helpful when evaluating infinite series like the Taylor series. To apply the Ratio Test, calculate the limit of the ratio of successive terms in the series as \( n \) approaches infinity.

The specific process to apply the Ratio Test involves calculating:

• The ratio of \( a_{n+1} \) (the \( n+1 \)th term) over \( a_n \) (the \( n \)th term).
• Taking the limit of this ratio as \( n \) goes to infinity.

If this limit is less than 1, the series converges absolutely. For our specific Taylor series of \( f(x) = b^x \), applying the Ratio Test revealed that the limit goes to zero. Zero is definitely less than 1, confirming that the series converges for all \( x \). Understanding the application of the Ratio Test offers insight into why certain series have certain convergence behaviors, reinforcing our ability to approximate functions with series confidently.