An alternating series 
converges if 
is monotonic and 
as 
. Here 
.
Let
. Then 
, which is positive for all 
, so 
is monotonically increasing for . This would mean 
must be a monotonically decreasing sequence over the same interval, and so must 
.
Because is monotonically increasing, but will still always be positive, it follows that as .
So, converges.
Let
Because
So,