In the alternating harmonic series, suppose the goal is to arrange the terms to get a new series that converges to -1/2. Start the new arrangement with the first negative term, which is -1/2. Whenever you have a sum that is less than or equal to -1/2, start introducing positive terms, taken in order, until the new total is greater than -1/2. Then add negative terms until the total is less than or equal to -1/2 again. Continue this process until your partial sums have been above the target at least three times and finish at or below it. If s_n is the sum of the first n terms of your new series, plot the points (n, s_n) to illustrate how the sums are behaving.