A252848 Decimal expansion of Sum_{n>0} Sum_{k=0..n} exp(k)/n! = e*(e^e - 1)/(e - 1).

2, 2, 3, 9, 1, 7, 1, 3, 1, 6, 8, 9, 4, 0, 2, 1, 7, 1, 1, 4, 4, 1, 3, 7, 7, 6, 9, 6, 3, 8, 8, 1, 7, 3, 1, 3, 2, 4, 9, 9, 2, 1, 6, 7, 2, 3, 3, 4, 4, 5, 1, 9, 4, 4, 7, 2, 8, 5, 1, 0, 9, 9, 1, 7, 2, 7, 9, 0, 6, 8, 9, 3, 4, 4, 9, 9, 8, 0, 5, 6, 1, 0

Offset: 2

Author

Richard R. Forberg, Dec 22 2014

Keywords

• nonn
• cons

Comments

Terms in the sum begin: 1 + (1 + e)/1 + (1 + e + e^2)/2 + (1 + e + e^2 + e^3)/6 + ... .

The largest term in the sum is at n = 2, where that term is 5.5536... .

The double sum converges to a similar algebraic form using any base for exponentiation. For instance, using Pi as the base shows the general closed form:

Pi*(e^Pi - e)/(Pi - 1), which equals 29.9584963... .

As the base approaches 1, the ratio converges to 2e = Sum_{n>0} Sum_{k=0..n} 1/n! = 5.43656... . See A019762.

Examples

			22.391713168940217114413... .
		

Programs

Mathematica

Sum[N[Sum[Exp[k]/n!, {k, 0, n}], 100], {n, 0, Infinity}]

Formula

e*(e^e - 1)/(e - 1).