In the Taylor series, what is the role of the remainder term?
A. Represents the error in approximation
B. Represents the integral of the function
C. Represents the derivative of the function
D. Represents the constant term
Answer (1)
In the Taylor series, the remainder term plays a crucial role. The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. The general form for the Taylor series of a function f ( x ) centered at a is:
f ( x ) = f ( a ) + f ′ ( a ) ( x − a ) + 2 ! f ′′ ( a ) ( x − a ) 2 + 3 ! f ′′′ ( a ) ( x − a ) 3 + … + R n ( x )
Here, R n ( x ) is the remainder term, also known as the error term. The role of the remainder term is critical because:
Represents the error in approximation: The correct answer is (A). The remainder term R n ( x ) highlights the difference between the actual function f ( x ) and its n -th degree polynomial approximation. This means that R n ( x ) tells us how far off our approximation is from the true value of the function.
Approximation quality: As n , the degree of the Taylor polynomial, increases, the remainder term R n ( x ) typically decreases in magnitude for functions that are well-approximated by polynomials. This implies that higher-degree Taylor polynomials usually provide better approximations near the point a .
Influence of function behavior: The accuracy of a Taylor series approximation and the size of the remainder term depend on both the function's behavior and how far x is from the center a .
Therefore, understanding the role of the remainder term is essential in determining the effectiveness and accuracy of Taylor series approximations.
