sum of reciprocals of Sylvester’s sequence

We will show that the sum of the reciprocals of the Sylvester numbers indeed converges to 1.

Let sn denote a partial sum of the series of reciprocals:


We would like to show that limnsn=1. Putting over a common denominator, we obtain


Define bn as follows:


Using this new definition and the definition of the Sylvester numbers, we can rewrite the expression for sn as follows:


Let us now consider this sequenceMathworldPlanetmath bn. We will start by deriving a recurrence relation:

bn+1-1 = j=0nij0i<n+1ai=i=0n-1ai+anj=0n-1ij0i<nai
= (an-1)+an(bn-1)

Simplifying, we have bn+1=anbn. Now, b2=1+a0+a1=6, hence we can solve the recursion with a productPlanetmathPlanetmath:

bn = b2i=2n-1ai
= b2a0a11=0n-1ai
= 1=0n-1ai
= an-1

Substituting this in the expression for sn yields


Since limnan=, it follows that limnsn=1.

Title sum of reciprocals of Sylvester’s sequence
Canonical name SumOfReciprocalsOfSylvestersSequence
Date of creation 2013-03-22 15:48:33
Last modified on 2013-03-22 15:48:33
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 7
Author rspuzio (6075)
Entry type Proof
Classification msc 11A55