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5.1.3. Show that (a₁ + a2 +. + an)² ≤ n (a² + a² + · Hint: Use the CBS inequality with x = α1 XX2 On and y + a²) for a₁ = R. CBS= Cauchy- Schwartz Inequality

5.1.3. Show that (a₁ + a2 +. + an)² ≤ n (a² + a² + · Hint: Use the CBS inequality with x = α1 XX2 On and y + a²) for a₁ = R. CBS= Cauchy- Schwartz Inequality

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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**5.1.3.** Show that \((\alpha_1 + \alpha_2 + \cdots + \alpha_n)^2 \leq n(\alpha_1^2 + \alpha_2^2 + \cdots + \alpha_n^2)\) for \(\alpha_i \in \mathbb{R}\). **Hint:** Use the CBS inequality with \(\mathbf{x} = \begin{pmatrix} \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_n \end{pmatrix}\) and \(\mathbf{y} = \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{pmatrix}\). *CBS = Cauchy-Schwartz Inequality*
Transcribed Image Text:**5.1.3.** Show that \((\alpha_1 + \alpha_2 + \cdots + \alpha_n)^2 \leq n(\alpha_1^2 + \alpha_2^2 + \cdots + \alpha_n^2)\) for \(\alpha_i \in \mathbb{R}\). **Hint:** Use the CBS inequality with \(\mathbf{x} = \begin{pmatrix} \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_n \end{pmatrix}\) and \(\mathbf{y} = \begin{pmatrix} 1 \\ 1 \\ \vdots \\ 1 \end{pmatrix}\). *CBS = Cauchy-Schwartz Inequality*
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