While working on my various and sundry Ï€ programs, I kept finding references to Gosper's paper Acceleration of Series, so I thought I'd find it on the web and have a read. It's quite the magnum opus of series acceleration with all sorts of gems that are, to be truthful, beyond my understanding. Worth reading, worth studying.
From the abstract:
The rate of convergence of infinite series can be accelerated b y a suitable splitting of each term into two parts and then combining the second part of the n-th term with the first part of the (n+1)-th term to get a new series and leaving the first part of the first term as an "orphan". Repeating this process an infinite number of times, the series will often approach zero, and we obtain the series of orphans, which may converge faster than the original series.
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