The Real And Complex Number Systems

which impliest that
lim sup |a n| 1/n |z| n = +∞.
n→∞
so, the series diverges. From above, we have proved the claim.
9.32 Given a power series ∑ a n x n whose coefficents are related by an
equation of the form
a n + Aa n−1 + Ba n−2 = 0 (n = 2, 3, ...).
Show that for any x for which the series converges, its sum is
a 0 + (a 1 + Aa 0 ) x
1 + Ax + Bx 2 .
Proof: Consider
which implies that
n=2
which implies that
n=0
∞∑
(a n + Aa n−1 + Ba n−2 ) x n = 0
n=2
∞∑
∞∑
a n x n + Ax a n−1 x n−1 + Bx 2
n=2
∞∑
∞∑
a n x n + Ax a n x n + Bx 2
which implies that
n=0
∞∑
n=0
∞
∑
n=0
∞
∑
n=2
a n x n = a 0 + (a 1 + Aa 0 ) x
1 + Ax + Bx 2 .
a n−2 x n−2 = 0
a n x n = a 0 + a 1 x + Aa 0 x
Remark: We prove that for any x for which the series converges, then
1 + Ax + Bx 2 ≠ 0 as follows.
Proof: Consider
( ) ∑
∞
1 + Ax + Bx
2
a n x n = a 0 + (a 1 + Aa 0 ) x,
n=0
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