EXAMPLE 55
EXAMPLE 56
Show how a series may be used to evaluate π.
SOLUTION: Since a series for tan−1 x may prove helpful. Note that
and that a series for is obtainable easily by long division to yield
If we integrate this series term by term and then evaluate the definite integral, we get
(Compare with series (5) in Common Maclaurin Series and note especially that this series
converges on −1 ≤ x ≤ 1.)
For x = 1 we have:
Here are some approximations for π using this series:
Since the series is alternating, the odd sums are greater, the even ones less, than the value of π. It is
clear that several hundred terms may be required to get even two-place accuracy. There are series
expressions for π that converge much more rapidly. (See Miscellaneous Free-Response Practice,
Problem 12.)
EXAMPLE 57