Problem 7 - Calculus Early Transcendentals

Question

In terms of the remainder, what does it mean for a Taylor series for a function \(f\) to converge to \(f ?\)

Step-by-Step Solution

Answer

Answer: In terms of the remainder, the convergence of a Taylor series to a function means that as the number of terms in the series increases, the remainder approaches zero for all points within the interval of convergence. Mathematically, this is expressed as: $$ \lim_{N\to\infty} R_N(x) = 0 $$

Step 1: Understanding Taylor Series

A Taylor series is a representation of a function as an infinite sum of its derivatives of all orders, evaluated at a specific point, usually around 0 (known as the Maclaurin series) or at some other point \(a\). The Taylor series of a function \(f\) around the point \(a\) is given by: $$ f(x) = \sum _{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n $$ Where \(f^{(n)}(a)\) is the n-th derivative of \(f\) evaluated at the point \(a\), and \(n!\) is the factorial of \(n\).

Step 2: Understanding the Remainder

The remainder of the Taylor series is the difference between the actual value of the function and the approximate value provided by the series up to a specific term \(N\). If we consider the sum of the series up to term \(N\), the remainder \(R_N(x)\) is given by: $$ R_N(x) = f(x) - \sum_{n=0}^{N} \frac{f^{(n)}(a)}{n!}(x-a)^n $$

Step 3: Understanding Convergence

For a Taylor series of a function \(f\) to converge to \(f\), the sum of the series must approach the actual value of the function as the number of terms increases. In other words, the remainder must become smaller and smaller as we take more terms into account. Mathematically, this means that for every point \(x\) in the interval of convergence, as \(N\) goes to infinity: $$ \lim_{N\to\infty} R_N(x) = 0 $$

Step 4: Putting It All Together

So, in terms of the remainder, for a Taylor series for a function \(f\) to converge to \(f\), it means that as we include more terms in the series, the remainder must approach zero for all \(x\) within the interval of convergence: $$ \lim_{N\to\infty} R_N(x) = \lim_{N\to\infty} \left[ f(x) - \sum_{n=0}^{N} \frac{f^{(n)}(a)}{n!}(x-a)^n \right] = 0 $$