1549901369-Elements_of_Real_Analysis__Denlinger_

8.6 Power Series 511

Writing the sum within braces in reverse order , this sum becomes
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L ak{xk-2(y-x) + xk-3(y2 x2) + xk-4(y 3 x3) + ... + xk-k(yk-1 _ xk-1).
k=2
Notice that each term in the sum within braces has y-x as a factor. Factoring
out y - x , the total sum above is

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L ak { xk-2(y - x) + xk- 3(y - x)(y + x) + xk-4(y - x)(y2 + yx + x2) + ...
k=2

+ xo(y _ x)(yk- 2 + yk-3x +. .. + yxk-3 + xk-2)}

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= (y - x) L ak { xk-2 + xk- 3(y + x) + x k-4(y2 + yx + x2) + ...

k=2

+ x o(yk-2 + yk-3x + ... + yxk-3 + xk-2)}.

Let M = max(lx l, IYI}. Then the absolute value of the above sum is

00

k=2

, 00

= IY - xi L laklMk-^2 {1+2 + 3 + .. · + (k - 1)}

k=2

I I

= y - x L..,, ~ I I ak (k - l)kMk-2

2

. (20)
k=2
Now M E 1° and t he series in (20) above, without absolute values, is the
t erm-by-term differentiation of the derived series of the given series at x = M.
Hence, by Theorem 8.6.11, this series must converge, say to S. Then , Vy E 1°,

I f(y~ = ~(x) - ~ akkxk-l I ~ IY -x lS.

Therefore, by the squeeze principle (Theorem 4.2.20 (b)),

lim f(y) - f(x) = f akkxk-^1.

y~x Y - X k=l

Part 2: Now consider the case c I= 0. Apply the chain rule to the result
of P art l. (Exercise 7 .) •

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Corollary 8.6.15 If f(x) = L ak(x - c)k has int erval of convergence I , with
k=O
nonempty interior, then

(a) f is continuous at every x in the interior of I.