514 Chapter 8 • Infinite Series of Real Numbers
(
Thus, tan-^1 x = x - x^3 5 7 )
3 + x^5 - x^7 + · · · +C. Letting x = 0, we find that
C = 0. Thus, for all x in (-1, 1),
(24)which must be the Maclaurin series for tan-^1 x. By Theorem 8.6.12, the radius
of convergence of this series is l.
1
( c) To derive a series representation of ( ) 2 we first note that
l+x
1 d 1
----
(l+x)2 dxl+x
Differentiating the series (21) term-by-term and multiplying by -1, we get(25)
= 1 - 2x + 3x^2 - 4x^3 + 5x^4 - · •..(d) Finally, the Maclaurin series for (l: x) 2 is easily found from (25)
using the algebra of limits:
= x - 2x^2 + 3x^3 - 4x^4 + 5x^5 - · · ·. D
BEHAVIOR AT ENDPOINTSWhen a function is representable as a power series everywhere in the interior
of its interval of convergence, the behavior of the power series at the endpoints
of this interval is unpredictable and must be investigated separately at each
endpoint. The power series may or may not converge at these endpoints, and
even when it converges, it may not converge to the value of the function at
that endpoint. Notice that the interval of convergence of the series (23) above
is (-1, 1) but the interval of convergence of the term-by-term integrated series
(24) above is [-1, l]. By Corollary 8.6.15 we can be sure that when -1<x<1,
the series (24) converges to tan-^1 x. But to what does the series (24) converge
when x = -1 or x = 1? We hope, of course, that it converges to tan -^1 x, but
we have not yet proved that.
The following theorem takes care of this problem. It assures us that if
the function is continuous at an endpoint and its power series representation
converges at that point, it will converge to the value of the function there. In
Example 8.6.21 we show how to apply this theorem.