Question

where the primes denote successive derivatives of \(f(x)\). (Depending on the function this series may converge for any increment \(\delta\) or only for values of \(\delta\) less than some nonzero "radius of convergence.") This theorem is enormously useful, especially for small values of \(\delta\), when the first one or two terms of the series are often an excellent approximation. \(^{11}\) (a) Find the Taylor series for \(\ln (1+\delta)\). (b) Do the same for cos \(\delta\). (c) Likewise sin \(\delta\). (d) And \(e^{\delta}\).

Short answer

(a) \( \ln(1+\delta) = \delta - \frac{\delta^2}{2} + \frac{\delta^3}{3} - \ldots \); (b) \( \cos \delta = 1 - \frac{\delta^2}{2} + \frac{\delta^4}{4!} - \ldots \); (c) \( \sin \delta = \delta - \frac{\delta^3}{3!} + \frac{\delta^5}{5!} - \ldots \); (d) \( e^{\delta} = 1 + \delta + \frac{\delta^2}{2!} + \ldots \)."

Step-by-step solution

Taylor Series Formula

The Taylor series for a function \( f(x) \) about \( x = a \) is given by \( f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots \). In these problems, we will consider \( a = 0 \) and express each function in terms of \( \delta \).

Taylor Series for \( \ln(1 + \delta) \)

The function \( f(x) = \ln(1 + x) \). Compute its derivatives: \( f'(x) = \frac{1}{1 + x} \), \( f''(x) = -\frac{1}{(1 + x)^2} \), \( f'''(x) = \frac{2}{(1 + x)^3} \). At \( x = 0 \), these are \( f(0) = 0 \), \( f'(0) = 1 \), \( f''(0) = -1 \), \( f'''(0) = 2 \), and so on. The Taylor series becomes \( \ln(1 + \delta) = \delta - \frac{\delta^2}{2} + \frac{\delta^3}{3} - \ldots \).

Taylor Series for \( \cos \delta \)

The function \( f(x) = \cos(x) \). Compute its derivatives: \( f'(x) = -\sin(x) \), \( f''(x) = -\cos(x) \), \( f'''(x) = \sin(x) \). At \( x = 0 \), these are \( f(0) = 1 \), \( f''(0) = -1 \), \( f''''(0) = 1 \), etc. The Taylor series becomes \( \cos(\delta) = 1 - \frac{\delta^2}{2} + \frac{\delta^4}{4!} - \ldots \).

Taylor Series for \( \sin \delta \)

The function \( f(x) = \sin(x) \). Compute its derivatives: \( f'(x) = \cos(x) \), \( f''(x) = -\sin(x) \), \( f'''(x) = -\cos(x) \). At \( x = 0 \), these are \( f(0) = 0 \), \( f'(0) = 1 \), \( f'''(0) = 0 \), \( f'''(0) = -1 \). The Taylor series becomes \( \sin(\delta) = \delta - \frac{\delta^3}{3!} + \frac{\delta^5}{5!} - \ldots \).

Taylor Series for \( e^{\delta} \)

The function \( f(x) = e^x \). All derivatives \( f'(x), f''(x), f'''(x) \), etc., are \( e^x \). At \( x = 0 \), \( f(0) = f'(0) = f''(0) = 1 \). The Taylor series becomes \( e^{\delta} = 1 + \delta + \frac{\delta^2}{2!} + \frac{\delta^3}{3!} + \ldots \).

Key concepts

Understanding Differentiation and the Taylor Series

Differentiation is a fundamental concept in calculus, used to find the rate at which a function is changing. In the context of a Taylor series, differentiation helps us derive the various terms of the series. A Taylor series approximates a function as an infinite sum of terms calculated from the values of its derivatives at a single point. This is especially useful because it allows us to express a complex function in simpler polynomial form.

To find a Taylor series for a function, you:

• Compute derivatives of the function at a particular point.
• Use those derivatives to form terms like \( f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \ldots \).

This series helps in calculating approximate values, simplifying complex expressions, and analyzing behavior near the expansion point.

Decoding the Radius of Convergence

The radius of convergence tells us the range of values for which a Taylor series converges to the actual value of the function. For any given function, the Taylor series may converge anywhere within this radius.

To understand this concept:

• Think of the radius of convergence as a limit beyond which the series may not provide a reliable approximation.
• It can be calculated using the ratio test, which involves the limit of the absolute values of the ratios of successive terms.

If a Taylor series has an infinite radius of convergence, it means the series converges for any value of the variable. Otherwise, it converges only within a specific interval centered at the expansion point.

Making Approximations with Taylor Series

Taylor series are powerful tools for approximating functions, especially when dealing with small values of the variable near the expansion point. These approximations make complex functions more manageable and easier to compute by focusing on just the first few terms of the series.

Approximation steps:

• Use the first term or first few terms of the Taylor series to get an estimated value.
• Check the accuracy of the approximation by considering the magnitude of the discarded higher-order terms.

For functions like \(\ln(1+\delta)\), \(\cos(\delta)\), and \(\sin(\delta)\), the first few terms can provide highly accurate results if \(\delta\) is small, making them extremely practical for real-world calculations and analyses.