It is well known that a polynomial
f
(
X
)
f(X)
over a commutative ring
R
R
with identity is nilpotent if and only if each coefficient of
f
(
X
)
f(X)
is nilpotent; and that
f
(
X
)
f(X)
is a zero divisor in
R
[
X
]
R[X]
if and only if
f
(
X
)
f(X)
is annihilated by a nonzero element of
R
R
. This paper considers the problem of determining when a power series
g
(
X
)
g(X)
over
R
R
is either nilpotent or a zero divisor in
R
[
[
X
]
]
R[[X]]
. If
R
R
is Noetherian, then
g
(
X
)
g(X)
is nilpotent if and only if each coefficient of
g
(
X
)
g(X)
is nilpotent; and
g
(
X
)
g(X)
is a zero divisor in
R
[
[
X
]
]
R[[X]]
if and only if
g
(
X
)
g(X)
is annihilated by a nonzero element of
R
R
. If
R
R
has positive characteristic, then
g
(
X
)
g(X)
is nilpotent if and only if each coefficient of
g
(
X
)
g(X)
is nilpotent and there is an upper bound on the orders of nilpotency of the coefficients of
g
(
X
)
g(X)
. Examples illustrate, however, that in general
g
(
X
)
g(X)
need not be nilpotent if there is an upper bound on the orders of nilpotency of the coefficients of
g
(
X
)
g(X)
, and that
g
(
X
)
g(X)
may be a zero divisor in
R
[
[
X
]
]
R[[X]]
while
g
(
X
)
g(X)
has a unit coefficient.