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(j) 5″ n! π

(j) 5″ n! π

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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Only part j please!

**Exercise: Monotonic Sequences** **Question:** Which of the following sequences are eventually monotone (or strictly) increasing (or decreasing)? Justify your answers, assuming the usual properties of trigonometric functions where necessary. **Sequences:** (a) \(\left\{\frac{(-1)^n}{n}\right\}\) (b) \(\left\{n - \frac{1}{n}\right\}\) (c) \(\left\{n + \left(\frac{(-1)^n}{n}\right)\right\}\) (d) \(\left\{n^2 - 10n + 100\right\}\) (e) \(\left\{2 + (-1)^n\right\}\) (f) \(\left\{\frac{3n^2 + (-1)^n}{n}\right\}\) (g) \(\left\{\sin \frac{n\pi}{2}\right\}\) (h) \(\left\{\sin n\pi\right\}\) (i) \(\left\{\frac{n}{2n}\right\}\) (j) \(\left\{\frac{5^n}{n!}\right\}\) (k) \(\left\{\cos \frac{\pi}{2n}\right\}\) (l) \(\left\{\sin \frac{\pi}{3n}\right\}\) (m) \(\left\{\frac{3n + 5}{n^2 - n - 2}\right\}\) (n) \(\left\{\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}}\right\}\)
Transcribed Image Text:**Exercise: Monotonic Sequences** **Question:** Which of the following sequences are eventually monotone (or strictly) increasing (or decreasing)? Justify your answers, assuming the usual properties of trigonometric functions where necessary. **Sequences:** (a) \(\left\{\frac{(-1)^n}{n}\right\}\) (b) \(\left\{n - \frac{1}{n}\right\}\) (c) \(\left\{n + \left(\frac{(-1)^n}{n}\right)\right\}\) (d) \(\left\{n^2 - 10n + 100\right\}\) (e) \(\left\{2 + (-1)^n\right\}\) (f) \(\left\{\frac{3n^2 + (-1)^n}{n}\right\}\) (g) \(\left\{\sin \frac{n\pi}{2}\right\}\) (h) \(\left\{\sin n\pi\right\}\) (i) \(\left\{\frac{n}{2n}\right\}\) (j) \(\left\{\frac{5^n}{n!}\right\}\) (k) \(\left\{\cos \frac{\pi}{2n}\right\}\) (l) \(\left\{\sin \frac{\pi}{3n}\right\}\) (m) \(\left\{\frac{3n + 5}{n^2 - n - 2}\right\}\) (n) \(\left\{\frac{1}{\sqrt{n}} - \frac{1}{\sqrt{n+1}}\right\}\)
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