Erdős–Borwein constant
The Erdős–Borwein constant, named after Paul Erdős and Peter Borwein, is the sum of the reciprocals of the Mersenne numbers. By definition it is: E = ∑ n = 1 ∞ 1 2 n − 1 ≈ 1.606695152415291763 … {\displaystyle E=\sum _{n=1}^{\infty }{\frac {1}{2^{n}-1}}\approx 1.606695152415291763\dots } == Equivalent forms == It can be proven that the following forms all sum to the same constant: E = ∑ n = 1 ∞ 1 2 n 2 2 n + 1 2 n − 1 {\displaystyle E=\sum _{n=1}^{\infty }{\frac {1}{2^{n^{2}}}}{\frac {2^{n}+1}{2^{n}-1}}} E = ∑ m = 1 ∞ ∑ n = 1 ∞ 1 2 m n {\displaystyle E=\sum _{m=1}^{\infty }\sum _{n=1}^{\infty }{\frac {1}{2^{mn}}}} E = 1 + ∑ n = 1 ∞ 1 2 n ( 2 n − 1 ) {\displaystyle E=1+\sum _{n=1}^{\infty }{\frac {1}{2^{n}(2^{n}-1)}}} E = ∑ n = 1 ∞ σ 0 ( n ) 2 n {\displaystyle E=\sum _{n=1}^{\infty }{\frac {\sigma _{0}(n)}{2^{n}}}} where σ0(n) = d(n) is the divisor function, a multiplicative function that equals the number of positive divisors of the number n.