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1. Prove that the sequence diverges but it has a convergent subsequence: (-1)" 2+(-1)", (a) (b) X = (rn) = (cos(na/2)) wwww. (c) X (n+(-1)"n).

1. Prove that the sequence diverges but it has a convergent subsequence: (-1)" 2+(-1)", (a) (b) X = (rn) = (cos(na/2)) wwww. (c) X (n+(-1)"n).

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.4: Fractional Expressions
Problem 65E
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Using basic real analysis 1, solve the question

**Problem 1:** Prove that the sequence diverges but it has a convergent subsequence: (a) \[ \left( \frac{(-1)^n}{2 + (-1)^n} \right) \] (b) \( X = (x_n) = (\cos(n\pi/2)) \) (c) \( X = (n + (-1)^{r_n}) \)
Transcribed Image Text:**Problem 1:** Prove that the sequence diverges but it has a convergent subsequence: (a) \[ \left( \frac{(-1)^n}{2 + (-1)^n} \right) \] (b) \( X = (x_n) = (\cos(n\pi/2)) \) (c) \( X = (n + (-1)^{r_n}) \)
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