Hello I have the series k*e^(-k^2) from one to infinity. By the integral test I find that the series diverges since the integral diverges to minus infinity (did the integral by substitution). By ratio test I reached that the a(k+1)/a(k) = (1+1/k) * e^(-2k-1) which tends to 0<1 as k tends to infinity so it converges by ratio test. What's happening here? I checked that I can use the integral test since my function is continuous, positive and non-increasing. With the integral test i reached the point -1/2 * [e^u] (with -1 as lower limit after the integration and + infinity as the upper limit after the integration) And finally I had -1/2 * infinity = - infinity so the improper integral doesn't exist since it is not finite and so our series diverges. Can anyone help me finding what am I doing wrong? Thanks in advance!!