In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm:
ln
(
1
+
x
)
=
x
−
x
2
2
+
x
3
3
−
x
4
4
+
⋯
In summation notation,
ln
(
1
+
x
)
=
∑
n
=
1
∞
(
−
1
)
n
+
1
n
x
n
.
The series converges to the natural logarithm (shifted by 1) whenever
−
1
<
x
≤
1
.
The series was discovered independently by Nicholas Mercator, Isaac Newton and Gregory Saint-Vincent. It was first published by Mercator, in his 1668 treatise Logarithmotechnia.
The series can be obtained from Taylor's theorem, by inductively computing the nth derivative of
ln
(
x
)
at
x
=
1
, starting with
d
d
x
ln
(
x
)
=
1
x
.
Alternatively, one can start with the finite geometric series (
t
≠
−
1
)
1
−
t
+
t
2
−
⋯
+
(
−
t
)
n
−
1
=
1
−
(
−
t
)
n
1
+
t
which gives
1
1
+
t
=
1
−
t
+
t
2
−
⋯
+
(
−
t
)
n
−
1
+
(
−
t
)
n
1
+
t
.
It follows that
∫
0
x
d
t
1
+
t
=
∫
0
x
(
1
−
t
+
t
2
−
⋯
+
(
−
t
)
n
−
1
+
(
−
t
)
n
1
+
t
)
d
t
and by termwise integration,
ln
(
1
+
x
)
=
x
−
x
2
2
+
x
3
3
−
⋯
+
(
−
1
)
n
−
1
x
n
n
+
(
−
1
)
n
∫
0
x
t
n
1
+
t
d
t
.
If
−
1
<
x
≤
1
, the remainder term tends to 0 as
n
→
∞
.
This expression may be integrated iteratively k more times to yield
−
x
A
k
(
x
)
+
B
k
(
x
)
ln
(
1
+
x
)
=
∑
n
=
1
∞
(
−
1
)
n
−
1
x
n
+
k
n
(
n
+
1
)
⋯
(
n
+
k
)
,
where
A
k
(
x
)
=
1
k
!
∑
m
=
0
k
(
k
m
)
x
m
∑
l
=
1
k
−
m
(
−
x
)
l
−
1
l
and
B
k
(
x
)
=
1
k
!
(
1
+
x
)
k
are polynomials in x.
Setting
x
=
1
in the Mercator series yields the alternating harmonic series
∑
k
=
1
∞
(
−
1
)
k
+
1
k
=
ln
(
2
)
.
The complex power series
∑
n
=
1
∞
z
n
n
=
z
+
z
2
2
+
z
3
3
+
z
4
4
+
⋯
is the Taylor series for
−
log
(
1
−
z
)
, where log denotes the principal branch of the complex logarithm. This series converges precisely for all complex number
|
z
|
≤
1
,
z
≠
1
. In fact, as seen by the ratio test, it has radius of convergence equal to 1, therefore converges absolutely on every disk B(0, r) with radius r < 1. Moreover, it converges uniformly on every nibbled disk
B
(
0
,
1
)
¯
∖
B
(
1
,
δ
)
, with δ > 0. This follows at once from the algebraic identity:
(
1
−
z
)
∑
n
=
1
m
z
n
n
=
z
−
∑
n
=
2
m
z
n
n
(
n
−
1
)
−
z
m
+
1
m
,
observing that the right-hand side is uniformly convergent on the whole closed unit disk.
