The Real And Complex Number Systems

9.38 For each real t, define f t (x) = xe xt / (e x − 1) if x ∈ R, x ≠ 0,
f t (0) = 1.
(a) Show that there is a disk B (0; δ) in which f t is represented by a power
series in x.
Proof: First, we note that ex −1
= ∑ ∞ x n
x n=0
:= p (x) , then p (0) = 1 ≠
(n+1)!
0. So, by Theorem 9. 26, there exists a disk B (0; δ) in which the reciprocal
of p has a power series exapnsion of the form
∞
1
p (x) = ∑
q n x n .
So, as x ∈ B (0; δ) by Theorem 9.24.
n=0
f t (x) = xe xt / (e x − 1)
( ∞
) (
∑ (xt) n ∞
)
∑ x n
=
n! (n + 1)!
n=0
n=0
∞∑
= r n (t) x n .
n=0
(b) Define P 0 (t) , P 1 (t) , P 2 (t) , ..., by the equation
and use the identity
f t (x) =
n=0
∞∑
n=0
P n (t) xn
, if x ∈ B (0; δ) ,
n!
∞∑
P n (t) xn
n! = ∑ ∞ etx P n (0) xn
n!
to prove that P n (t) = ∑ n
k=0 (n k ) P k (0) t n−k .
Proof: Since
f t (x) =
∞∑
n=0
n=0
P n (t) xn
n! = etx x
e x − 1 ,
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