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
BC ONLY(Compare with series (5) on the previous pages and note especially that this
series converges on −1 ≤ x ≤ 1.) For x = 1 we have:
Then
and
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 __
Use a series to evaluate to four decimal places.
SOLUTION: Although cannot be expressed in terms of elementary
functions, we can write a series for eu, replace u by (−x^2 ), and integrate term by
term. Thus,
so