Assuming an = ln (n+2)/[(n+ 1) x], it is easy to know that it is a positive sequence.

Because the harmonic series is divergent and ln (n+2) >: 1, when x

X > will be discussed below; 1

For the sake of understanding, simple and clear x & gt=3 cases.

Because when n is large enough, there is a

So x & gt=3 converges.

When x is (1, 3)

Firstly, a lemma is proved: for an arbitrarily small t>0, when n is large enough, there is LN (n+2).

This should be easier.

Then an < (n+ 1) (x-t), obviously an converges to (1, 3).

So An> 1 converges.