I need some help determining whether or not the equation in the attachment diverges or converges. I'm trying to teach myself the integral test, and cannot seem to get this one to work. Any help would be much appreciated.
Lets look at the integral
Now we can see that the first and last terms are
and respectively.
Now let us consider
If we use integration by parts once we get
Now if we let
and
So this would gives us
Now by letting and respectively we come up with
Now making the appropriate backsubs we get
Putting this all back together we get
So putting this completely back together and remembering that we get
Can you go from there?
I know that this is beating a dead horse, but I seem to still have a problem. The integral test keeps telling me the series converges, but the graph of the equation is basically telling me that it diverges. Am I just messing myself up on this one, or did I make a mistake with the integral test?