Sequence monotonic homework problem help

[sonutulsiani]

Homework Statement

Consider the sequence.

http://www.webassign.net/cgi-bin/symimage.cgi?expr=a_n = 4 n + 1/n

(a) Determine whether the sequence is increasing, decreasing, or not monotonic.

(b) Is the sequence bounded?

Homework Equations

The Attempt at a Solution

I got the answer to a and that is increasing

I am not sure about (b). How do I find that out??

 

[rekshaw]

Bounded means, it will reach a point where it will stop increasing more and more quickly. For example, if you had just 1/x^2, it will get smaller and smaller and smaller until the increment is tiny and the sum of the sequence will end up converging to a number.

your sequence is 4n + 1/n ... therefore, as n gets bigger and bigger, what will happen? try it out with increasingly big numbers to see the trend. if it keeps getting bigger faster and faster, then its UNBOUNDED.

 

[Mark44]

Bounded means, it will reach a point where it will stop increasing more and more quickly. For example, if you had just 1/x^2, it will get smaller and smaller and smaller until the increment is tiny and the sum of the sequence will end up converging to a number.

A sequence {s_n} is bounded if there are numbers M and N such that M <= s_n <= N for all n = 1, 2, 3, ...

 

[sonutulsiani]

Yes it does keep on increasing. It doesn't stop. But I am not sure if it's faster and faster or not. I know it keeps getting bigger. So it's unbounded? And what about (a) increasing?

 

[Mark44]

You have a_n = 4n + 1/n, which has no upper bound. A sequence is increasing if a_{n + 1} >= a_n for all n >= 1. (Some texts use the phrase strictly increasing if a_{n + 1} > a_n.)