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K Find the remainder in the Taylor series centered at the point a for the following function. Then show that lim R₁(x) = 0 for all x in the interval of n-∞ convergence. f(x) = ex. a=0 Find a bound for R₁(x) that does not depend on c, and thus holds for all n. Choose the correct answer below. elx| OA. R₂(x)(n+1)! |x/n+1 elx OC. R₂(x)² (n+1)! -/x/n+1 E. R₂(x) ≤ (n+1)! *** ex S x+1 ex OB. R₁(x) ² (n+1)! 1 OD. Rn(x)(n+1)! |x-e (x-e)n + 1 1 OF. Rn(x) ≤ (n+1)! (x-e)n +1

K Find the remainder in the Taylor series centered at the point a for the following function. Then show that lim R₁(x) = 0 for all x in the interval of n-∞ convergence. f(x) = ex. a=0 Find a bound for R₁(x) that does not depend on c, and thus holds for all n. Choose the correct answer below. elx| OA. R₂(x)(n+1)! |x/n+1 elx OC. R₂(x)² (n+1)! -/x/n+1 E. R₂(x) ≤ (n+1)! *** ex S x+1 ex OB. R₁(x) ² (n+1)! 1 OD. Rn(x)(n+1)! |x-e (x-e)n + 1 1 OF. Rn(x) ≤ (n+1)! (x-e)n +1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Transcription for Educational Website **Topic: Understanding the Remainder in Taylor Series** **Objective:** Find the remainder in the Taylor series centered at the point \( a \) for the following function, and show that \( \lim_{{n \to \infty}} |R_n(x)| = 0 \) for all \( x \) in the interval of convergence. **Function Details:** - Function: \( f(x) = e^{-x} \) - Center: \( a = 0 \) --- **Problem Statement:** Find a bound for \( |R_n(x)| \) that does not depend on \( c \), and thus holds for all \( n \). Choose the correct answer from the options below. **Options:** - **A.** \( |R_n(x)| \leq \frac{e^{|x|}}{(n+1)!} |x|^{n+1} \) - **B.** \( |R_n(x)| \geq \frac{e^{-x}}{(n+1)!} |x - e| \) - **C.** \( |R_n(x)| \geq \frac{e^{|x|}}{(n+1)!} |x|^{n+1} \) - **D.** \( |R_n(x)| \geq \frac{1}{(n+1)!} (x-e)^{n+1} \) - **E.** \( |R_n(x)| \leq \frac{e^x}{(n+1)!} x^{n+1} \) - **F.** \( |R_n(x)| \leq \frac{1}{(n+1)!} (x-e)^{n+1} \) --- **Instructions:** Analyze the options carefully and select the one that correctly bounds the remainder \( |R_n(x)| \) for the given function. Understanding the nature of the exponential function and the properties of Taylor series will be essential in solving this problem.
Transcribed Image Text:### Transcription for Educational Website **Topic: Understanding the Remainder in Taylor Series** **Objective:** Find the remainder in the Taylor series centered at the point \( a \) for the following function, and show that \( \lim_{{n \to \infty}} |R_n(x)| = 0 \) for all \( x \) in the interval of convergence. **Function Details:** - Function: \( f(x) = e^{-x} \) - Center: \( a = 0 \) --- **Problem Statement:** Find a bound for \( |R_n(x)| \) that does not depend on \( c \), and thus holds for all \( n \). Choose the correct answer from the options below. **Options:** - **A.** \( |R_n(x)| \leq \frac{e^{|x|}}{(n+1)!} |x|^{n+1} \) - **B.** \( |R_n(x)| \geq \frac{e^{-x}}{(n+1)!} |x - e| \) - **C.** \( |R_n(x)| \geq \frac{e^{|x|}}{(n+1)!} |x|^{n+1} \) - **D.** \( |R_n(x)| \geq \frac{1}{(n+1)!} (x-e)^{n+1} \) - **E.** \( |R_n(x)| \leq \frac{e^x}{(n+1)!} x^{n+1} \) - **F.** \( |R_n(x)| \leq \frac{1}{(n+1)!} (x-e)^{n+1} \) --- **Instructions:** Analyze the options carefully and select the one that correctly bounds the remainder \( |R_n(x)| \) for the given function. Understanding the nature of the exponential function and the properties of Taylor series will be essential in solving this problem.
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