- #1

saddlepoint

- 6

- 0

## Homework Statement

Show that {n

^{2}- n + 5} is increasing and hence show that {x

_{n}} is convergent when

{x

_{n}} = exp[(3n

^{2}- 3n +14) / (n

^{2}- n + 5)]

You may assume exp x < exp y when x < y, but may not use any properties of the limit of exp x as x → 3.

## Homework Equations

## The Attempt at a Solution

I tried to show x

_{n+1}≥ x

_{n}to show {n

^{2}- n + 5} was increasing.

So I got (n+1)

^{2}- (n+1) + 5 ≥ n

^{2}- n + 5

Therefore n

^{2}+ n + 5 ≥ n

^{2}- n + 5 and so n ≥ -n

Does this show that {n

^{2}- n + 5} is increasing? Is it enough proof?

Going onto the next bit of the question. I assume because you can't use any properties of the limit of exp x as x → 3 that you must use the Sandwich Theorem? This being that when you have sequences such as:

{z

_{n}} ≤ {x

_{n}} ≤ {y

_{n}}

that if {z

_{n}} and {y

_{n}} converge, {x

_{n}} also must converge.

After this I'm not really too sure what to do, or whether this is actually the right way to approach this question.

Has anyone got any help that they could offer me please?