1 − 2 + 4 − 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs. As a geometric series, it is characterized by its first term, 1, and its common ratio, −2. As a series of real numbers it diverges, so in the usual sense it has no sum. In a much broader sense, the series has a generalized sum of . Gottfried Leibniz considered the divergent alternating series as early as 1673. He argued that by subtracting either on the left or on the right, one could produce
2-8-8-2 still used on steel mill lines and other railroads until 1983. Other equivalent classifications are: UIC classification: 1DD1 (also known as German classification and Italian classification) French classification: 140+041 Turkish classification: 45+45 Swiss classification: 4/5+4/5 The first 2-8-8-2 was built in 1909 by Baldwin, who sold two to the Southern Pacific Railroad (classified MC-1), and then three each to the Union Pacific Railroad and UP-owned Oregon Railroad and Navigation Company. Baldwin conceived the type as an expansion of the 2-6-6-2 permitting a greater tractive effort. The next order for the type was from the Southern Pacific; these differed in being