Originally Posted by

**ecMathGeek** Use Integration by Parts on this:

INT (udv) = uv - INT(vdu) ... I'm hoping you're familiar with this.

At the end, we will also need to use limits in order to integrate from 1 to infinity.

Let u = lnx <--> du = 1/xdx

Let dv = 1/x^2dx <--> v = -1/x

So, INT lnx/x^2 dx becomes:

-lnx/x - INT (-1/x^2)

-lnx/x - 1/x

-(lnx + 1)/x

Now, plugging in the limits of integration, using a limit for the upper limit of infinity:

lim[n->infinity] -(lnx + 1)/x from (1 to n)

lim[n->infinity] -(ln(n) + 1)/n + (ln(1) + 1)/1

Now we need to use L'Hopitals to simplify the limit

lim[n->infinity] (-1/x)/1 + 1 = 0/1 + 1 = 1