Section 2.1 Taylor’s Formula and The Remainder Estimation Theorem

Theorem 2.1. Taylor’s Formula.

Let \(f\) be a function that has \(n+1\) continuous derivatives on an open interval \(I\) containing \(a\text{.}\) Then for each \(x\in I\) and for each positive integer \(n\text{,}\) there exists a number \(c\) between \(a\) and \(x\) such that

\begin{equation*} f(x) = p_n(x) + R_n(x), \end{equation*}

where

\begin{equation*} p_n(x)= \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k, \end{equation*}

and

\begin{equation*} R_n(x) = \frac{f^{(n+1)}(c)}{(n+1)!} (x-a)^{n+1}. \end{equation*}

To see a visual representation of Taylor’s formula, watch the following animation.

Figure 2.2. Taylor’s Formula Animation

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Figure 2.3. Taylor’s Formula: \(f(x) = p_n(x) + R_n(x)\text{.}\) The Taylor polynomial \(p_n\) agrees with \(f\) at \(x = a\text{;}\) away from \(a\text{,}\) the vertical gap between them is the remainder \(R_n(x)\text{.}\)

Proof.

The proof of the remainder theorem is based on the mean value theorem and Taylor’s theorem and we have included the proof in Section 2.6. The proof is postponed to the end of this section because it requires some technical details that are not necessary for understanding the main ideas of this section. The students are not expected to know the proof, however, they should understand the statement and its implications. Interested students are encouraged to read the proof.

Remark 2.4. Finding an upper bound for the error term without knowing the value of \(c\).

Usually the value of \(c\) is not explicitly known. However, we may manage to find an upper bound for the error term \(R_n(x)\) without knowing the exact value of \(c\text{.}\) This is achieved by finding an upper bound \(M\) for \(|f^{(n+1)}(c)|\text{,}\) where \(c\) is between \(a\) and \(x\text{,}\) and then using this upper bound to estimate the error term.

Theorem 2.5. The Remainder Estimation Theorem.

\begin{equation*} |R_n(x)| \leq \frac{M}{(n+1)!} |x-a|^{n+1}, \end{equation*}

where \(M\) is an upper bound for \(|f^{(n+1)}(c)|\) on the interval between \(a\) and \(x\text{.}\)

Proof.

The proof of the remainder estimation theorem is based on the Taylor’s formula Theorem 2.1. Suppose that we can find an upper bound \(M\) for \(|f^{(n+1)}(c)|\) on the interval between \(a\) and \(x\text{,}\) i.e., \(|f^{(n+1)}(c)| \leq M\text{,}\) then:

\begin{equation*} |R_n(x)| = \left| \frac{f^{(n+1)}(c)}{(n+1)!} (x-a)^{n+1} \right| \leq \frac{M}{(n+1)!} |x-a|^{n+1}. \end{equation*}

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