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