WOLFRAM|DEMONSTRATIONS PROJECT

False Logarithm Series

​
a
2
In a paper titled "Consideration of Some Series Which Are Distinguished by Special Properties", published in 1753, Leonhard Euler considers
s
a
(x)=
∞
∑
n=1
1
1-
n
a
n-1
∏
k=0
(1-x
-k
a
)
. He proves that
s
a
(
n
a
)=n
for non-negative integers
n
, just like
log
a
(
n
a
)=n
, but shows that this function is different from
log
a
(x)
. This Demonstration compares the plots of
s
a
(
x
a
)
and
log
a
(
x
a
)
for different values of
a
. Red points correspond to pairs of integers on the line
y=x
.