Write out the complete proof of the integral remainder formula in Taylor's theorem.
Write out the complete proof of the integral remainder formula in Taylor's theorem.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Title: Complete Proof of the Integral Remainder Formula in Taylor’s Theorem**
*Introduction*
Taylor’s Theorem provides an approximation of a k-times differentiable function around a point using a Taylor polynomial and a remainder term. The integral form of the remainder provides insight into the accuracy of the approximation.
*Statement of the Theorem*
Let \( f \) be a function that is \( (n+1) \)-times differentiable on the closed interval \([a, b]\). The Taylor series expansion of \( f(x) \) around the point \( a \) is given by:
\[
f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^n(a)}{n!}(x-a)^n + R_n(x)
\]
Where the remainder \( R_n(x) \) in the integral form is expressed as:
\[
R_n(x) = \int_a^x \frac{f^{(n+1)}(t)(x-t)^n}{n!} \, dt
\]
*Proof of the Integral Remainder Formula*
1. **Setup**:
Consider the function \( F(t) = \frac{f^{(n+1)}(t)(x-t)^n}{n!} \). Evaluate \( F(t) \) over the interval \([a, x]\).
2. **Differentiation under the integral sign**:
Show that the nth derivative of the Taylor polynomial \( P_n(x) \) effectively cancels out all terms except for the remainder when evaluated from \( a \) to \( x \).
3. **Properties of integration**:
Use the Fundamental Theorem of Calculus to link derivatives of an integral to the integrand \( F(t) \).
4. **Conclude with the integral form**:
Demonstrate that \( R_n(x) = \int_a^x F(t) \, dt \) and simplify to match the formula given for \( R_n(x) \).
*Conclusion*
The integral form of the remainder in Taylor's Theorem not only represents the difference between the actual function value and the polynomial approximation but also ensures accuracy by incorporating higher derivatives in the evaluation over the interval.
*Note*: This abstract form depends on the function's differentiability](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85f1beef-b59a-44ac-a29d-5ba6518650e4%2F987966d9-2992-4dfb-b597-8f945b51ddf2%2Fl2vo02o_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Complete Proof of the Integral Remainder Formula in Taylor’s Theorem**
*Introduction*
Taylor’s Theorem provides an approximation of a k-times differentiable function around a point using a Taylor polynomial and a remainder term. The integral form of the remainder provides insight into the accuracy of the approximation.
*Statement of the Theorem*
Let \( f \) be a function that is \( (n+1) \)-times differentiable on the closed interval \([a, b]\). The Taylor series expansion of \( f(x) \) around the point \( a \) is given by:
\[
f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^n(a)}{n!}(x-a)^n + R_n(x)
\]
Where the remainder \( R_n(x) \) in the integral form is expressed as:
\[
R_n(x) = \int_a^x \frac{f^{(n+1)}(t)(x-t)^n}{n!} \, dt
\]
*Proof of the Integral Remainder Formula*
1. **Setup**:
Consider the function \( F(t) = \frac{f^{(n+1)}(t)(x-t)^n}{n!} \). Evaluate \( F(t) \) over the interval \([a, x]\).
2. **Differentiation under the integral sign**:
Show that the nth derivative of the Taylor polynomial \( P_n(x) \) effectively cancels out all terms except for the remainder when evaluated from \( a \) to \( x \).
3. **Properties of integration**:
Use the Fundamental Theorem of Calculus to link derivatives of an integral to the integrand \( F(t) \).
4. **Conclude with the integral form**:
Demonstrate that \( R_n(x) = \int_a^x F(t) \, dt \) and simplify to match the formula given for \( R_n(x) \).
*Conclusion*
The integral form of the remainder in Taylor's Theorem not only represents the difference between the actual function value and the polynomial approximation but also ensures accuracy by incorporating higher derivatives in the evaluation over the interval.
*Note*: This abstract form depends on the function's differentiability
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