and r = −2, so the sum is . Euler summation In his 1755 Institutiones, Leonhard Euler effectively took what is now called the Euler transform of , arriving at the convergent series . Since the latter sums to , Euler concluded that . His ideas on infinite series do not quite follow the modern approach; today one says that is Euler summable and that its Euler sum is . The Euler transform begins with the sequence of positive terms: a0 = 1, a1 = 2, a2 = 4, a3 = 8,... The sequence of forward differences is then Δa0 = a1 − a0 = 2 − 1 = 1, Δa1 = a2 − a1 = 4 − 2 = 2, Δa2 = a3 − a2 = 8 − 4 = 4, Δa3 = a4 − a3 = 16 − 8 = 8,... which is just the same sequence. Hence the iterated forward difference sequences all start with for every n. The Euler transform is the series This is a convergent geometric series whose sum is by the usual formula. Borel summation The Borel sum of is also ; when Émile Borel introduced the limit formulation of Borel summation in 1896, this was one of his first examples after 1 − 1 + 1 − 1 + ⋯ -adic numbers The sequence of partial sums associated with in the 2-adic metric is and when expressed in base 2 using two's complement, and the limit of this sequence is in the 2-adic metric. Thus . See also 1 + 2 + 4 + 8 + ⋯ Notes References Divergent series Geometric series P-adic numbers