Appendix: Accelerating convergence for other series
We can easily play the same game with the other series mentioned in this article. For example take the Gregory/Leibnitz series. There is a subtlety with this case as this is an alternating series which, as you add it up, takes values which oscillate above and below the value of To avoid this we will define the sum by
This sum is always slightly greater than . Then
As before we will assume that and that
Comparing the two expressions for and we get
Again, as before, we can find the values of the terms by expanding out each of these fractions and comparing powers of when is large. This is slightly more complicated than last time an we must use the result that for large values of
Doing this we find that
so that
As before we will compare the approximations to the sum given by and by the corrections to given by
and





1 
0.8666666666666 
0.741666666666667 
0.804166666666667 
0.784635416666667 
5 
0.8080789523513 
0.783078952351398 
0.785578952351398 
0.785397702351398 
10 
0.7972961955693 
0.784796195569399 
0.785421195569399 
0.785398148694399 
Again if we compare these values to the true value of then we see excellent agreement, with real economy of effort. In fact which is a far better estimate for than the one that we obtained earlier by adding up 100 terms of the same series.