Answered: Decide whether or not the given…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Question

Decide whether or not the given sequence converges to a limit. If it does not, find, in each case, at least one convergent subsequence. We suppose n= 1, 2, 3,...

x_n = (-1)ⁿ(1-(1/n))

Expert Solution

Step 1: step 1

To determine whether the given sequence $x subscript n equals not stretchy left parenthesis negative 1 not stretchy right parenthesis to the power of n not stretchy left parenthesis 1 minus 1 over n not stretchy right parenthesis$ converges to a limit or not, we can analyze its behavior.

Let's consider the sequence for different values of n:

- For n = 1: $x subscript 1 equals not stretchy left parenthesis negative 1 not stretchy right parenthesis to the power of 1 not stretchy left parenthesis 1 minus 1 over 1 not stretchy right parenthesis equals negative 1 not stretchy left parenthesis 1 minus 1 not stretchy right parenthesis equals negative 1$
- For n = 2: $x subscript 2 equals not stretchy left parenthesis negative 1 not stretchy right parenthesis squared not stretchy left parenthesis 1 minus 1 half not stretchy right parenthesis equals 1 not stretchy left parenthesis 1 minus 0.5 not stretchy right parenthesis equals 0.5$
- For n = 3: $x subscript 3 equals not stretchy left parenthesis negative 1 not stretchy right parenthesis cubed not stretchy left parenthesis 1 minus 1 third not stretchy right parenthesis equals negative 1 not stretchy left parenthesis 1 minus 0.3333 not stretchy right parenthesis almost equal to negative 0.6667$
- For n = 4: $x subscript 4 equals not stretchy left parenthesis negative 1 not stretchy right parenthesis to the power of 4 not stretchy left parenthesis 1 minus 1 fourth not stretchy right parenthesis equals 1 not stretchy left parenthesis 1 minus 0.25 not stretchy right parenthesis equals 0.75$

It appears that the sequence alternates between positive and negative values and that the absolute values of the terms are approaching 0 as n increases. However, the sequence does not converge to a single limit because it oscillates between positive and negative values indefinitely.