1. The problem statement, all variables and given/known data Determine whether the following series converges absolutely, converges conditionally or diverges. Show your work in applying any tests used. sigma[k=1,inf] [(-1)^k*k/sqrt(k^4+2)] 2. Relevant equations integral csc(x) dx = -ln|sec(x)+cot(x)| + C csc(x) = 1/sin(x) sec(x) = 1/cos(x) cot(x)= 1/tan(x) tan(x) = opposite/adjacent sin(x) = opposite/hypotenuse cos(x) = adjacent/hypotenuse d[cot(x)]/dx=-csc(x)^2 x^a*x^b = x^(a+b) x^a/x^b = x^(a-b) 3. The attempt at a solution I tried using the divergence test and applied Lpitals rule (or however you spell it) multiple times and just gave up so I then went on to tried to proceed using the integral test. My issue came when I was trying to take the integral of k/sqrt(k^4+2). I got -1/2 ln|sqrt(k^4+2)/sqrt(2) + k^2/sqrt(2)|. I don't see what I'm doing wrong. I took the derivative of this using wolfram alpha, http://www.wolframalpha.com/input/?i=d%5B-1%2F2ln|sqrt%28k^4%2B2%29%2Fsqrt%282%29%2Bk^2%2Fsqrt%282%29|%5D%2Fdk and got -k/sqrt(k^4+2), which isn't equal to my original integrand, k/sqrt(k^4+2). Apparently the correct antiderivative of the integral is 1/2ln|sqrt(k^4+2)/sqrt(2)+k^2/sqrt(2)|, as you can see by checking with wolfram alpha http://www.wolframalpha.com/input/?i=d%5B1%2F2ln|sqrt%28k^4%2B2%29%2Fsqrt%282%29%2Bk^2%2Fsqrt%282%29|%5D%2Fdk, it's correct. So apparently I'm not suppose to have a negative in front of my antiderivative but I don't see were I went wrong, I guess I was suppose to have -1/2 integral csc(theta) dtheta, which would of given me the correct antiderivative but I don't see why I'm suppose to have a negative sign out there, when I substituted k and dk the +/- and +/- simplified to just +, I think, which doesn't need to be shown because something is generally assumed to be positive if there's not a negative sign in front of. This however is the only place that I can think I made a error, in simplifying +/- * +/-, as this is the only point in the evaluation that I messed around with positive and negative signs. This leads me to believe that +/-*+/- can be simplified to - and not +, but this is incorrect, but I don't see were else I could of gone wrong with positive and negative signs. I don't see what I'm doing wrong. Thanks for any help! Let me know if you can't follow my work, apparently it's all right except for the negative symbol in my antiderivative that's not suppose to be there. See the attachment. Click on it to view it in a window that pops up. If you want to view it at a larger scale click on it again to view it in a new tab blown up some.