    Next: Bibliography Up: 12. Power Series Previous: 12.8 Proof of the   Index

# 12.9 Some XVIII-th Century Calculations

The following proofs that and use XVIII-th century standards or rigor. You should decide what parts are justified. I denote by below. By the geometric series formula, If where , , then so By equating the real and imaginary parts, we get For , this yields Thus, , so Hence, Since two antiderivatives of a function differ by a constant for some constant . When , we get so and thus (12.79)

For , this gives us (which is the Gregory-Leibniz-Madhava formula). We can rewrite (12.79) as Again, since two antiderivatives of a function differ by a constant, there is a constant such that For , this says and for , this says Subtract the second equation from the first to get i.e., and thus (12.80)

Let Subtract (12.80) from this to get Hence, , and then . An argument similar to the following was given by Jacob Bernoulli in 1689 [31, p 443]. Let Then Subtract the second series from the first to get Therefore, 12.81   Exercise. a) Explain why Bernoulli's argument is not valid.
b) Give a valid argument proving that 12.82   Note. The notation was introduced by William Jones in 1706 to represent the ratio of the circumference to the diameter of a circle[15, vol2, p9]. Both Maple and Mathematica designate by Pi .

The notation was introduced by Euler in 1727 or 1728 to denote the base of natural logarithms[15, vol 2, p 13]. In Mathematica is denoted by E . In the current version of Maple there is no special name for ; it is denoted by exp(1) .    Next: Bibliography Up: 12. Power Series Previous: 12.8 Proof of the   Index