Lemma 1.3
(Partial Summation)
.
Assuming that:
(
a
n
)
n
∈
ℕ
are complex numbers
x
≥
y
≥
0
f
:
[
y
,
x
]
→
ℂ
is continuously differentiable
Then
∑
y
<
n
≤
x
a
n
f
(
n
)
=
A
(
x
)
f
(
x
)
−
A
(
y
)
f
(
y
)
−
∫
y
x
A
(
t
)
f
′
(
t
)
d
t
,
where for
t
≥
1
, we define
A
(
t
)
=
∑
n
≤
t
a
n
=
∑
n
=
1
⌊
t
⌋
a
n
.