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