In response to: Sum of Pascal's triangle reciprocals

Approved additions to these Online Encyclopedia of Integer Sequences (OEIS):# HermannSW

## Re: Sum of Pascal's triangle reciprocals |

## Sum of Pascal's triangle reciprocals
Many binomial identities are known, and Pascal's triangle is well known, too.
I asked myself back in 1983 at school: What is the sum of the reciprocals of Pascal's triangle? Of course the sum is infinite because of the 1's on left and right border. Of course the sum is infinite again because of the harmonic series on left and right border. And the answer was (and is) 3/2 ! So based on the Theorem further below the Corollary (sum being 3/2) can be prooven pretty easily. And here is the main Theorem, a nice decomposition of each unit fraction into binomial coefficient reciprocals. Starting summation of binomial coefficient reciprocals at row j+1 for column j gives 1/(j-1).
My proof is by mathematical induction, two A4 pages long and will not be posted here. It is left as exercise to the reader ;-) So, why is this posting marked with XSLT tag? Easier to proof it the theorem validity can just be "seen" by actually doing the summations.
Hermann. |