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Putnam
1990 Putnam
B2
P_n(n,x): given a product, prove a sum
P_n(n,x): given a product, prove a sum
Source: Putnam 1990 B2
July 12, 2013
induction
Putnam
college contests
Problem Statement
Prove that for
∣
x
∣
<
1
|x| < 1
∣
x
∣
<
1
,
∣
z
∣
>
1
|z| > 1
∣
z
∣
>
1
,
1
+
∑
j
=
1
∞
(
1
+
x
j
)
P
j
=
0
,
1 + \displaystyle\sum_{j=1}^{\infty} \left( 1 + x^j \right) P_j = 0,
1
+
j
=
1
∑
∞
(
1
+
x
j
)
P
j
=
0
,
where
P
j
P_j
P
j
is
(
1
−
z
)
(
1
−
z
x
)
(
1
−
z
x
2
)
⋯
(
1
−
z
x
j
−
1
)
(
z
−
x
)
(
z
−
x
2
)
(
z
−
x
3
)
⋯
(
z
−
x
j
)
.
\dfrac {(1-z)(1-zx)(1-zx^2) \cdots (1-zx^{j-1})}{(z-x)(z-x^2)(z-x^3)\cdots(z-x^j)}.
(
z
−
x
)
(
z
−
x
2
)
(
z
−
x
3
)
⋯
(
z
−
x
j
)
(
1
−
z
)
(
1
−
z
x
)
(
1
−
z
x
2
)
⋯
(
1
−
z
x
j
−
1
)
.
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