1549901369-Elements_of_Real_Analysis__Denlinger_

8.7 Analytic Functions 523

where (~) = 1 and (~) = a(a - l)(a -^2 1l· .. (a - k + l) when k ;::: 1, is

called the binomial series.

We shall show that this series converges to (1 + x)°' for all Ix I < l. But first

we prove a lemma that we will need.

Lemma 8. 7.9 \:/ lxl < 1, and \fa E JR, lim n ( a) lxln = 0.

n->oo n

n

Proof. Suppose lxl < l. Apply the ratio test to the series L k(~)xk:

k=O

lim I (k + l)(k~l)xk+l I= lxl lim (k + 1) lim I (k~1) I

k->oo k (~) xk k->oo k k->oo (~)

. la(a-l)(a-2)···(a-k) kl I
= lxl }_:.1! (k + l)l · -a-(a---1)-(a---2)- .-.. -(a __ k+l)

= lxl lim I ak - k I = lxl lim I ~ -; I = !xi < l.

k->oo + 1 k->oo 1 + k

So by t he ratio test, this series converges. Therefore, its general term must have
limit 0. •

Theorem 8.7.10 Given an arbitrary real number a,\:/ x E (-1,1),

00

(1 + x)°' = L (~)xk.

k=O

(Convergence at the endpoints -l and 1 depends on the value of a.)^11

Proof. Let f(x) = (l+x)°', where a is an arbitrary but fixed real number.
We find the Maclaurin series for f.

f(x) = (1 + x)°' f(O) = 1

J'(x) = a(l + x)°'-^1 J'(O) = a

f"(x) = a(a - 1)(1 + x)^0 -^2 f"(O) = a(a -1)

J(k)(x) = a(a - 1) ···(a - k + 1)(1 + x)°'-k

= kl(~) (1 + x )°'-k J(k) (0) = kl(~).

11. For more complete details, see pages 567 - 572 of [32].