1549901369-Elements_of_Real_Analysis__Denlinger_

528 Chapter 8 • Infinite Series of Real Numbers

00

f(x) = L ak[(d - c) + (x - d)]k and, using the binomial theorem,

k=O

= L^00 ak [ L k (k). (d- c)k-J(x-d).^1 ·]

k=O j=O J

= L^00 [ L k ak (k). (d- c)k-J(x-d)J. ·].

k=O j=O J

(34)

We first prove that this series converges absolutely in an open interval
centered at d. Choose any x such that Ix - di < p - le - di, and let t
Ix - di+ le - d i. Then 0 < t < p, and

f lakl ft (~)(d-c)k-j(x-d)JI :S: f lakl (t (~)ld-clk-jlx-dlj)

k=O j = O J k=O j=O J

= f lakl [Id - cl+ Ix - dl]k (binomial theorem)

k=O

00

= L lak ltk, which converges since 0 < t < p.

k=O

c-p c d x c +p

Fig ure 8.4

T hus the series (34) converges absolutely whenever Ix - di < p - le - di,

and represents the sum of all terms in the "infinite matrix" below, adding first

across the rows and then adding the row sums:

ao (~)^0 0 0

a1(~)(d-c) a1G)(x -d) 0 0

az(~)(d-c)^2 az(i)(d -c)(x - d) az (;) ( x -d)^2 0

a3(~)(d-c)^3 a3(i)(d- c)^2 (x - d) a3(~)(d - c)(x - d)^2 a3m(x -d)^3

In Theorem 8.7.16 below,^12 we prove that we get the same sum by adding
down the columns and then adding the column sums. Now the sum of column

12. We place Theorem 8.7.16 below to avoid interupting the flow of ideas here.