Prove that if x_n --> infinity, then the sequence given by
x_n / x_n+1 is convergent.
I am supposed to use the definition of divergence, so I know there is a number n for which all x_n > M for any M chosen. i think you are using an incorrect definition of divergence
to support what i have written in red:
define as .
then is divergent since its not convergent. But if we take (say) we can't find any which would satisfy .
Also can you please explicitly write what definition of you are using??
The definition of a limit at infinity from my book is as follows:
sn → ∞ as n → ∞
provided that for every number M there is an integer N so that
sn ≥ M whenever n ≥ N.
That definition is standard. Now apply it to the example I gave you.
It will show that .