Whats wrong with the harmonic series?
Notice that this series has one term 1 that is at least 1, and one that is at least (namely ), two others that are at least ( and ), four others that are at least ( through ), others that are at least ( through ), and so on. Each of these groups sums to at least , so we get an endless number of ‘s which must diverge. Suppose, for simplicity, that the terms in your series are arranged in decreasing order. (If not rearrange them.) You can make a histogram, creating blocks, each of width 1 and height between j – 1 and j given by the j-th term in your series. Then the value of the series is the area between the x axis and the top line of this histogram, between 0 and infinity. You can then, with reasonable luck, be able to define a nice smooth curve that meet the left topmost points of each of your blocks, and another (the same one moved a distance one to the right) that meets the right topmost points of each of your blocks. In the case of the harmonic series, these are the curves and . Wh
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