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Mathematics LibreTexts

7.6: Taylor's Theorem Revisited

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    22683
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    The following is a version of Taylor's Theorem with an alternative form of the remainder term.

    Theorem \(\PageIndex{1}\)

    (Taylor's Theorem)

    Suppose \(f \in C^{(n+1)}(a, b), \alpha \in(a, b),\) and

    \[P_{n}(x)=\sum_{k=0}^{n} \frac{f^{(k)}(\alpha)}{k !}(x-\alpha)^{k}.\]

    Then, for any \(x \in(a, b)\),

    \[f(x)=P_{n}(x)+\int_{\alpha}^{x} \frac{f^{(n+1)}(t)}{n !}(x-t)^{n} d t.\]

    Proof

    By the Fundamental Theorem of Calculus, we have

    \[\int_{\alpha}^{x} f^{\prime}(t) d t=f(x)-f(\alpha),\]

    which implies that

    \[f(x)=f(\alpha)+\int_{\alpha}^{x} f^{\prime}(t) d t.\]

    Hence the theorem holds for \(n=0 .\) Suppose the result holds for \(n=k-1,\) that is,

    \[f(x)=P_{k-1}(x)+\int_{\alpha}^{x} \frac{f^{(k)}(t)}{(k-1) !}(x-t)^{k-1} d t.\]

    Let

    \[F(t)=f^{(k)}(t),\]

    \[g(t)=\frac{(x-t)^{k-1}}{(k-1) !},\]

    and

    \[G(t)=-\frac{(x-t)^{k}}{k !}.\]

    Then

    \[\begin{aligned} \int_{\alpha}^{x} \frac{f^{(k)}(t)}{(k-1) !}(x-t)^{k-1} d t &=\int_{\alpha}^{x} F(t) g(t) d t \\ &=F(x) G(x)-F(\alpha) G(\alpha)-\int_{\alpha}^{x} F^{\prime}(t) G(t) d t \\ &=\frac{f^{(k)}(\alpha)(x-\alpha)^{k}}{k !}+\int_{\alpha}^{x} \frac{f^{(k+1)}(t)}{k !}(x-t)^{k} d t, \end{aligned}\]

    Hence

    \[f(x)=P_{k}(x)+\int_{\alpha}^{x} \frac{f^{(k+1)}(t)}{k !}(x-t)^{k} d t,\]

    and so the theorem holds for \(n=k\). \(\quad\) Q.E.D.

    Exercise \(\PageIndex{1}\)

    (Cauchy form of the remainder)

    Under the conditions of Taylor's Theorem as just stated, show that

    \[\int_{\alpha}^{x} \frac{f^{(n+1)}(t)}{n !}(x-t)^{n} d t=\frac{f^{(n+1)}(\gamma)}{n !}(x-\gamma)^{n}(x-\alpha)\]

    for some \(\gamma\) between \(\alpha\) and \(x .\)

    Exercise \(\PageIndex{2}\)

    (Lagrange form of the remainder)

    Under the conditions of Taylor's Theorem as just stated, show that

    \[\int_{\alpha}^{x} \frac{f^{(n+1)}(t)}{n !}(x-t)^{n} d t=\frac{f^{(n+1)}(\gamma)}{(n+1) !}(x-\alpha)^{n+1}\]

    for some \(\gamma\) between \(\alpha\) and \(x .\) Note that this is the form of the remainder in Theorem \(6.6 .1,\) although under slightly more restrictive assumptions.

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