Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.


f(x) = 1/(1 - x), a=0

Verified step by step guidance

1. Identify the function and the center of the Taylor polynomial: here, the function is \(f(x) = \frac{1}{1 - x}\) and the Taylor polynomial is centered at \(a = 0\).
2. Recall the formula for the remainder (Lagrange form) of the nth-order Taylor polynomial centered at \(a\): \[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x - a)^{n+1}\] where \(c\) is some value between \(a\) and \(x\).
3. Find the \((n+1)\)th derivative of \(f(x)\). Since \(f(x) = (1 - x)^{-1}\), use the general formula for derivatives of this form: \[f^{(k)}(x) = k! \cdot (1 - x)^{-(k+1)}\] with appropriate sign considerations.
4. Substitute \(a = 0\) into the remainder formula and express \(R_n(x)\) in terms of \(f^{(n+1)}(c)\) and \((x - 0)^{n+1} = x^{n+1}\).
5. Write the final expression for the remainder \(R_n(x)\) as: \[R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} x^{n+1}\] where \(c\) lies between \$0\( and \)x\(, and \)f^{(n+1)}(c)\( is the \)(n+1)\(th derivative evaluated at \)c$.

Key Concepts

Taylor Polynomial

A Taylor polynomial of order n approximates a function near a point a using a finite sum of derivatives of the function at a. It is expressed as the sum of terms involving the function's derivatives up to the nth order, each multiplied by powers of (x - a) and divided by factorials.

Remainder (Error) Term in Taylor Series

The remainder term Rₙ represents the difference between the actual function value and its nth-order Taylor polynomial approximation. It quantifies the error and can be expressed using forms like Lagrange's form, involving the (n+1)th derivative evaluated at some point between a and x.

Derivatives of f(x) = 1/(1 - x)

For f(x) = 1/(1 - x), the nth derivative is n!/(1 - x)^(n+1). Understanding this pattern is essential to formulating the Taylor polynomial and remainder term, as the derivatives determine the coefficients and the behavior of the remainder.