510 Chapter 8 • Infinite Series of Real Numbers
Proof. Exercise 6. 0In the next group of results we show that a function representable as a
power series with a positive radius of convergence must be quite well-behaved
in its interval of convergence: it must be differentiable and, consequently, contin-
uous and integrable there. We shall see that these results follow simply and el-
egantly from one fundamental result, Theorem 8.6. 14. Unfortunately, the proof
of this theorem is somewhat tedious. In Chapter 9 it will be seen as a straight-
forward consequence of "uniform convergence,'' to be defined in that chapter.
Theorem 8.6.14 If a function f is representable as a power series with nonzero
radius of convergence, then f is differentiable at every point in the interior of
its interval of convergence; moreover, its derived series is its derivative. That
is, if
00
f(x) = L ak(x - c)k
k=Owith interval of convergence I , then at every point x in the interior of J,
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J'(x) = L akk(x - c)k-l _
k=l
Proof. Part 1: We first consider the case c = 0 to minimize the notational
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complexity. Suppose f ( x) = L akxk for every x E J^0 , the interior of the
k=O
interval of convergence of this power series. Let x be a fixed member of I°.
Then, Vy E J^0 , y =f. x,
00 00 00
f(y)-f(x) = L akyk-L akxk = L ak(yk- xk)
k=O k=O k=l
(When k = 0, the term is 0.)
So,
f(y) - f(x) - ~ L..,, ak k x k-1 = L..,, ~ ak {yk - xk - k x k-1}
y - x k=l k=2 y - x00
(When k = 1, the term is 0.)
L ak { (yk-1 + yk-2x + yk-3x2 + ... + yxk-2 + xk-1) _ kxk-1}
k=2
00
L ak{(yk-1 _ xk-1) + (yk-2x _ xk-1) + (yk-3x2 _ xk-1) + ...
k=2
+ (y2xk-3 _ xk-1) + (yxk-2 _ xk-1) + (xk-1 _ xk-1)}.