-- Arithmetic: sum of i for i = 1..n, against n(n+1)/2 --
  n =     5  brute =       15  closed =       15  MATCH
  n =    10  brute =       55  closed =       55  MATCH
  n =   100  brute =     5050  closed =     5050  MATCH
  n =  1000  brute =   500500  closed =   500500  MATCH

-- Geometric with ratio a = 1/2 (convergent): sum a^i, i = 0..n --
  n =   0  sum = 1.000000   limit 1/(1-a) = 2.000000
  n =   3  sum = 1.875000   limit 1/(1-a) = 2.000000
  n =  10  sum = 1.999023   limit 1/(1-a) = 2.000000
  n =  30  sum = 2.000000   limit 1/(1-a) = 2.000000

-- Geometric with ratio a = 2 (divergent): sum 2^i, i = 0..n --
  n =   3  sum =    15.000000   2^(n+1)-1 = 15.000000
  n =   5  sum =    63.000000   2^(n+1)-1 = 63.000000
  n =  10  sum =  2047.000000   2^(n+1)-1 = 2047.000000
  n =  20  sum = 2097151.000000   2^(n+1)-1 = 2097151.000000

-- Harmonic: H(n) = sum 1/i, against ln n + 0.5772156649 --
  n =        1  H(n) = 1.000000   ln n + gamma = 0.577216
  n =      100  H(n) = 5.187378   ln n + gamma = 5.182386
  n =    10000  H(n) = 9.787606   ln n + gamma = 9.787556
  n =  1000000  H(n) = 14.392727   ln n + gamma = 14.392726

-- Telescoping: sum 1/(i(i+1)) for i = 1..n equals 1 - 1/(n+1) --
  n =      1  sum = 0.500000   1 - 1/(n+1) = 0.500000
  n =      5  sum = 0.833333   1 - 1/(n+1) = 0.833333
  n =     50  sum = 0.980392   1 - 1/(n+1) = 0.980392
  n =   5000  sum = 0.999800   1 - 1/(n+1) = 0.999800
