Just a and b Ucqucice (CXn}?Suppose that xn → ∞ and yn → -∞. Find examples of such sequences with thegiven property.a) The sequence {Xn + Yn} converges to 00.b) The sequence {Xn + Yn} converges to -0o.c) The sequence {Xn + Yn} converges to a, where a is a real number.d) The sequence {Xn + yn} is bounded but does not converge.

Answered: e that xn → ∞ and yn → -00. Find…

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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Question

Just a and b

Ucqucice (CXn}?
Suppose that xn → ∞ and yn → -∞. Find examples of such sequences with the
given property.
a) The sequence {Xn + Yn} converges to 00.
b) The sequence {Xn + Yn} converges to -0o.
c) The sequence {Xn + Yn} converges to a, where a is a real number.
d) The sequence {Xn + yn} is bounded but does not converge.

Expert Solution

Step 1

(a) Consider the sequence

$\{x_{n}\} = n^{2} + n \{y_{n}\} = - n$

Then

$\begin{array}{rcl} \lim_{n \to \infty} x_{n} & = & \lim_{n \to \infty} (n^{2} + n) \\ = & \lim_{n \to \infty} n^{2} + \lim_{n \to \infty} n \\ = & \infty + \infty \\ = & \infty \\ \lim_{n \to \infty} y_{n} & = & \lim_{n \to \infty} (- n) \\ = & - \lim_{n \to \infty} n \\ = & - \infty \end{array}$

Consider

$\begin{array}{rcl} \lim_{n \to \infty} (x_{n} + y_{n}) & = & \lim_{n \to \infty} (n^{2} + n - n) \\ = & \lim_{n \to \infty} n^{2} \\ = & \infty \end{array}$