4-8-4+4-8-4 classes of 4-8-4+4-8-4 steam locomotives across the globe, all of which were constructed by Beyer, Peacock and Company, the owners of the Garratt patent. The predecessor 4-8-2+2-8-4 Double Mountain was probably the optimal Garratt wheel arrangement, with the four-wheeled leading bogies and the two-wheeled trailing trucks on each engine unit ensuring stability at speed and with sixteen coupled wheels for traction. More coupled wheels would inhibit the locomotive on tight curves, while the only advantage of more non-coupled wheels, such as on the Double Northern, was to reduce the axle loading. The second Double Northern locomotive class were the AD60
1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ In mathematics, the infinite series is an elementary example of a geometric series that converges absolutely. There are many different expressions that can be shown to be equivalent to the problem, such as the form: 2 + 2 + 2 + ... The sum of this series can be denoted in summation notation as: As with any infinite series, the infinite sum is defined to mean the limit of the sum of the first terms as approaches infinity. Multiplying by 2 reveals a useful relationship: Subtracting from both sides, As approaches