Absolute Convergence: Solve w/ Supercomputer in Scientific Notation

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ArcainineFalls531
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Determine wheter the sum from n=2 to infinity of ((-1)^(n+1))/(ln(n)) converges absolutely, converges conditionally, or diverges. Also assume you have a supercomputer that can add 10^15 terms per second (which is very fast for even a supercomputer). If you wanted to estimate the sum to within an error of .01, how long would this take? Give your answer in years. preferably in scientific notation. In what state will you find the Earth when your computer has completed this computation? :bugeye:

When I first started trying this problem, I attempted using the Power series, as it's something we've been recently covering in class. Today we went over the derivatives and antiderivatives of functions such as this, and their relation to each other. The major place where I'm running into trouble is with the imaginary supercomputer thing. Also, I'm not sure if I'm starting in the right place? I appreciate any help received. Thanks.
 

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converges conditionally. compare with the series 1/n not sure about the part about the supercomputer
 
You should have a simple way of bounding the 'tail' of the series (as it's an alternating one). Using this bound, find out how many terms you have to add to get within the specified error.
 
In a convergent alternating series, the error is always less than the absolute value of the first term of the tail. Have you learned about the alternating series test?