RePoSS #11: The Mathematics of Niels Henrik Abel: Continuation ...

232 Chapter 12. ABEL’s reading of CAUCHY’s new rigor and the binomial theorem
CAUCHY’s structure ABEL’s structure
Definition of convergence
✟
✟❍
✟✟✙
❍❍❍❥
Cauchy Geometric
criterion progression
❄
Root test
❄
Ratio test
Definition of convergence
❄
Cauchy criterion
❄
Ratio test
Figure 12.1: Comparison of CAUCHY’s and ABEL’s structures of the basic theory of
infinite series.
An auxiliary theorem: Lehrsatz III. As his third theorem, 26 ABEL presented an aux-
iliary result which — although not difficult — was put to great use in the proofs to
follow. He demonstrated that if {tn} denoted a sequence whose partial sums were
bounded,
m
∑ tk < δ for all m ∈ N,
k=0
and {εn} denoted a decreasing sequence of positve terms, then
rm =
m
∑ εktk < δε0 for all m ∈ N.
k=0
ABEL’S proof consisted of a rather simple manipulation, in which he observed that
with
each term could be written as
and, thus,
rm =
=
m
∑ εktk =
k=0
Since {εn} was decreasing,
26 (N. H. Abel, 1826f, 314)
pm =
m
∑ tk, k=0
t k = p k − p k−1,
m
∑ εk (pk − pk−1) =
k=0
m−1
∑ pk (εk − εk+1) + εmpm.
k=0
0 < ε k − ε k+1 < ε k < ε0,
m
∑ εkp k −
k=0
m−1
∑ εk+1p k
k=0