Question

Using the integral test, how do you show whether `sum ln(n)/(n)^2` diverges or converges from n=1 to infinity?

Answer 1

See the explanation.

Explanation:

I assume that the difficulty is in integrating

`int lnx/x^2 dx = int x^-2 lnx dx`

Use parts: `u = lnx` and `dv = x^-2 dx`

So we get

`int x^-2 lnx dx = -lnx/x+intx^-2 dx`

`= -lnx/x - 1/x = -(lnx+1)/x`

Or perhaps the difficulty is in evaluating

`lim_(brarroo) [-(lnb+1)/b + (ln1 + 1)/1] = lim_(brarroo) [-(lnb+1)/b +1]`

Use l'Hopital's Rule to get `lim_(brarroo) [-(lnb+1)/b] = 0`