# two easy questions

• Aug 29th 2011, 08:26 AM
boromir
two easy questions
Here's something I thought would be easy to prove (if I'm corret of course)

If $\displaystyle x_n$ -> x and x >a, then $\displaystyle x_n>a$ for infinitly many values of n.

My working: we need $\displaystyle |x_n -x|<|x-a|= x-a$ for n sufficently large

->$\displaystyle |x_n|<|x|+x-a$

whatever I do I can't seem to deduce x_n > a for all but finite n

My second question is very simple. In the ratio test, if the limit does not exist what does limit superior greater than 1 imply?

Thanks.
• Aug 29th 2011, 08:52 AM
Plato
Re: two easy questions
Quote:

Originally Posted by boromir
Here's something I thought would be easy to prove (if I'm corret of course)
If $\displaystyle x_n$ -> x and x >a, then $\displaystyle x_n>a$ for infinitly many values of n.

Let $\displaystyle \varepsilon = \frac{{x - a}}{2} > 0$

If $\displaystyle |x_n-x|<\varepsilon$ then $\displaystyle a<\frac{a+x}{2}<x_n$.
• Aug 29th 2011, 05:12 PM
vincisonfire
Re: two easy questions
Quote:

Originally Posted by boromir
My second question is very simple. In the ratio test, if the limit does not exist what does limit superior greater than 1 imply?

If the inequality $\displaystyle {a_{n+1}\over a_n} > 1$ holds for all but finitely many n, then the series diverges. You can't say anything about limsup i think