Question

Using the integral test, how do you show whether `n/(n^2+1)` diverges or converges?

Answer 1

`int_1^oo x/(x^2+1) dx`

Let `u=x^2+1`, so that `du= 2x dx`

`int x/(x^2+1) dx = 1/2 int (2x)/(x^2+1) dx = 1/2 1/u du = 1/2 ln abs(u) +C`

`int_1^oo x/(x^2+1) dx =lim_(brarr oo) 1/2ln(x^2+1)|_1^b` which diverges,

So the series diverges.