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The Riemann series theorem (or Riemann rearrangement theorem) loosely states that for any conditionally convergent series, and any [extended] real number, there exists a rearrangement of the series that converges to the given real number. The proof of this beautiful theorem given in Rudin's book on analysis gives an algorithm for constructing the desired rearrangement. Here, I implement this algorithm and plot the progression of the partial sums of the rearranged series using a few options for conditionally convergent series. Note that for real numbers larger than (more negative than) 8 (-8) the simulation can take a very long time to converge.