The powers of a function f: A → A are defined recursively by
(a) Using calculus, or by other means, find the sum 1 + 2x + 3x2 + 4x3 + ⋯ + nxn−1.
(b) Assuming |x| < 1, find 1 + 2x + 3x2 + 4x3 + ⋯.
(c) Assuming |x| < 1, find 1 + 3x + 5x2 + 7x3 + ⋯.
(d) Assuming |x| < 1, find 1 +3x + 6x2 + 9x3 + 12x4 + ⋯.
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