Consider the functions C: R → R and S: R → R defined by C(x) = = Σ n=0 n 2n (-1)"x² (2n)! and S(x) = Σ n=0 n 2n+1 (-1)"x² (2n+1)! Show that there is a least positive number x_ such that C(x) = 0 Assume C(x) > 0 for all x > 0. Show that S is increasing, C is decreasing and concave down and derive a contradiction.
Consider the functions C: R → R and S: R → R defined by C(x) = = Σ n=0 n 2n (-1)"x² (2n)! and S(x) = Σ n=0 n 2n+1 (-1)"x² (2n+1)! Show that there is a least positive number x_ such that C(x) = 0 Assume C(x) > 0 for all x > 0. Show that S is increasing, C is decreasing and concave down and derive a contradiction.
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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![Consider the functions \( C: \mathbb{R} \to \mathbb{R} \) and \( S: \mathbb{R} \to \mathbb{R} \) defined by
\[
C(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \quad \text{and} \quad S(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}
\]
Show that there is a least positive number \( x_0 \) such that \( C(x_0) = 0 \).
Assume \( C(x) > 0 \) for all \( x > 0 \). Show that \( S \) is increasing, \( C \) is decreasing and concave down and derive a contradiction.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F37e8ed93-7bef-4409-89ed-52264f64a27e%2Fa650ff21-3cce-43e7-bd46-8888070b0168%2F81nzy4_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the functions \( C: \mathbb{R} \to \mathbb{R} \) and \( S: \mathbb{R} \to \mathbb{R} \) defined by
\[
C(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \quad \text{and} \quad S(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}
\]
Show that there is a least positive number \( x_0 \) such that \( C(x_0) = 0 \).
Assume \( C(x) > 0 \) for all \( x > 0 \). Show that \( S \) is increasing, \( C \) is decreasing and concave down and derive a contradiction.
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