91Ó°ÊÓ

The Taylor series is a way to represent a function as an infinite sum of terms. Each of these terms is calculated by taking derivatives of the function at a single point. This series helps us express complicated functions in simpler polynomial forms, which are much easier to work with.

For a function \( f(x) \), the Taylor series centered at a point \( a \) is:\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \ldots\]

• \( f(a) \) is the function value at \( a \)
• \( f'(a) \), \( f''(a) \), etc., are derivatives at \( a \)
• \( (x-a) \) represents the distance from the center point \( a \)

When you expand \( \ln(5-x) \), you perform similar steps by centering the series around a simple logarithmic form. The process involves transforming the original function into a known series, like \( \ln(1-x) \), and then making a substitution to find the new series expansion. This substitution helps tailor the series to fit the original function's form.

Radius of convergence is essential when working with power series. It tells us the interval within which the series will accurately represent the function. Any \( x \) falling inside this radius ensures the series converges to the actual value of the function.

The radius of convergence \( R \) can be found using the ratio test or by determining how transformations affect the series:

• For \( \ln(1-x) \), the radius \( R = 1 \) means that the series converges for \( |x| < 1 \).
• If we substitute \( u = \frac{x}{5} \) for \( \ln(5-x) \), it transforms the problem as \( |\frac{x}{5}| < 1 \), resulting in \( |x| < 5 \).

So, by multiplying the input by 5, the radius stretches to include more values of \( x \), giving a wide range where the series accurately reflects the natural logarithm function.

The natural logarithm, denoted as \( \ln(x) \), is a special logarithm with base \( e \), where \( e \) is approximately 2.71828. It is frequently encountered in calculus because it simplifies many mathematical expressions involving growth rates and time.

• \( \ln(x) \) is only defined for \( x > 0 \)
• The logarithmic growth is slow compared to polynomial or exponential growth

The natural logarithm has a simple derivative: \( \frac{d}{dx} [ \ln(x) ] = \frac{1}{x} \). This simplicity makes it an appealing choice for integration and differentiation tasks because transformations involving \( \ln(x) \) often simplify solutions.

When expanding \( \ln(5-x) \) into a power series, we transform it into the form \( \ln(1-u) \). This transformation allows us to use the naturally occurring series properties to decode the complexity behind \( \ln(5-x) \) and express it as an understandable polynomial form. This method preserves the basic properties of \( \ln \) while making computations more tractable.