Question

How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?

Accepted Answer

Recall that the Taylor polynomial of degree n for a function $$ f $$ centered at $$ a  is $$given by the formula:

$$ P_n(x) = \sum_{k=0}^n \frac{f^{(k)}(a)}{k!} (x - a)^k $$

where $$ f^{(k)}(a) $$ denotes the $$ k -th $$derivative of $$ f $$ evaluated at $$ a . $$Understand that the Taylor series of $$ f $$ centered at $$ a  is $$the infinite sum of all these terms, expressed as:

$$ T(x) = \sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!} (x - a)^k $$

This series represents the function $$ f  as an $$infinite polynomial expansion around $$ a . $$Recognize that each Taylor polynomial $$ P_n(x)  is $$essentially the partial sum of the Taylor series up to degree $$ n . In $$other words, the Taylor polynomial approximates $$ f  by $$truncating the infinite series after $$ n $$ terms. Note that as $$ n $$ increases, the Taylor polynomial $$ P_n(x) $$ generally provides a better approximation to $$ f(x) $$ near the point $$ a $$, and the Taylor series is the limit of these polynomials as $$ n 	o \infty $$, assuming the series converges to $$ f(x) . $$Therefore, the Taylor polynomials centered at $$ a $$ are the finite-degree approximations that build up to the full Taylor series centered at $$ a $$, which is the infinite-degree polynomial representation of $$ f $$ around that point.

How are the Taylor polynomials for a function f centered at a related to the Taylor series of the function f centered at a?

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Concetti chiave

Taylor Polynomial

Taylor Series

Relationship Between Taylor Polynomials and Taylor Series