468 Trigonometric functions
This and the sandwich theorem imply that 0 as n and the defini.
tion involving (5.622) shows that
(8) tan'x
=x-x3+-3- 5-x7 I x.9-x.ll
7 9 11
when 0 < x < 1. Changing the sign of x changes the signs of both sides and
shows that the formula is correct when -1 < x 5 1. Putting x = 1 gives the
famous Leibniz formula(9) 7r4=1-3+5-7+9-11I
which is of particular interest to those who have not previously seen a formula
from which it could be calculated. Putting x = 1 in (6) shows that we must
take many terms of the series (8) to obtain a sum agreeing with 7r/4 to three or
four decimal places, and we say that the series "converges slowly." The series
in (8) converges more rapidly when x is nearer 0. We are not now in the comput-
ing business, but it is easy to verify that(10) 4=tan'i+tan13'
by taking tangents of both sides, and to use (8) to show that ir is roughly 3.14.
The formula of Machin (1680-1751)(11) 4 = 4 tan-'^1 - tan' 1391
which is not so easily proved, is used by professional computers.(^13) Find whether the function f for which f(x) = x + sin x has an inverse
and, if so, whether the inverse is differentiable.
(^14) Let B be an interval of values of x; it could be the interval x > 0, and it
could be the interval -1 5 x < 1. Let D be the operator or transformer or
differentiator which, when applied to a function f which is defined and differenti-
able over E, produces the function 0 for which 4(x) = f(x) when x is jn E.
The domain A of D is then the set of functions f that are defined and differentiable
over B, and the range R of D is the set of functions q6 that are defined over E
and are derivatives of functions differentiable over E. Show that D does not
possess an inverse.
15 Let E be the interval -1 < x < 1. Let D, be the restriction of the oper-
ator D of the preceding problem to the domain 0, consisting of functions f
defined over E for which f (O) = 0 and f exists and is continuousover E. Show
that D3 has an inverse.
(^16) Let E be the interval -1 < x < 1. Let I, be the operator or transformer
or integrator which, when applied to a function g which is continuous over E,
produces the function G for which
G(x) = fox g(t) dt
when x is in E. Describe the domain and range of I, and show that I, hasan
inverse.