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

12.7. From power series to absolute convergence 249
ABEL’s Lehrsatz V derived from DU BOIS-REYMOND’s theorem Actually, ABEL’S
fifth theorem is a consequence of DU BOIS-REYMOND’S theorem.
Corollary 1 Assume that
∞
∑ vn (x) δ
n=1
n
(12.20)
is convergent for some δ > 0 and that the functions vn are continuous functions of x on some
interval I. Then, for any 0 < α < δ, the function
f (x) =
∞
∑ vn (x) α
n=1
n
(12.21)
is a continuous function of x on the interval I. ✷
PROOF We wish to use DU BOIS-REYMOND’S theorem to prove the theorem stated
above. For this, we write
and denote
f (x) =
∞
∑ vn (x) δ
n=1
n � α
�n δ
wn (x) = vn (x) δ n and
�
α
�n µn = .
δ
Now, we are ready to test the requirements of DU BOIS-REYMOND’S theorem. The
first requirement, that ∑ µn converges absolutely is obviously satisfied since α < δ.
Secondly, the functions wn are obviously finite and continuous since this was required
of vn. Lastly, we have to show that limn→∞ wn (x) is also finite. However, this is
an easy consequence to draw from the convergence of (12.20) which ensures us that
limn→∞ wn (x) = 0 for all x ∈ I. Thus, the continuity of (12.21) follows from DU
BOIS-REYMOND’S theorem. �
Box 4: ABEL’s Lehrsatz V derived from DU BOIS-REYMOND’s theorem