An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

226

Integration of an infinite series.

Bk. III. ch. IV.

xl
Jfn (X) dx = x n (X1)
XO
Now first of all, the series of these integral functions cannot possibly
converge unless:
Xi
Lim pn (x) == Lim fn(x) dx
xo
vanish for n = oo. We cannot infer that this condition is fulfilled
of itself because
Xi
fn (x) d = (x, - Xo) f (Xo + O (X - Xo))
x0
and Lim fn = 0. For instance:
xi
Lim x ne-'xzdx -== Lim (1 — e-~xl) - 1
0
and is not equal to
x Lim (n xe- ~I x 2) = 0.
When the series formed of the integral functions of the several
terms is a continuous function of x, and its derived series for every
value in the interval from x0 to x1 is equal to the series F(x), it
expresses the integral of the original series.
According to the first Theorem of last ~ this requires the series of
the integral functions to have the property that for any number 6 however
small, a place n can be found such that for it and all higher values
the derived of the remainder term BRn(x) shall be less than 6.
But by the second Theorem it is a sufficient condition, that the
given series converges uniformly.
This second theorem can be seen directly as follows: If
_F(x) =, f(x+) () + f3 x) +. fn-l () + Pn (X),
and for the entire interval from x, to xl a single n can be found
for which and for all greater values the continuous function P (x)
shall remain less in amount than Y, we shall have:
Pn (x)dx -- (x1 - x) Pn (xo + Ox1 - Xo), therefore
Xo
Xi
J F(x) adx == q (x, I) + T2 (x ) +.. Pn-1(X) + (X~-Xo) P' (Xo + Oi x - x,).
10 o
If now n be arbitrarily increased, we have
J '(x) cdx = pI(x1) -- +P2 (x) -* e.- 1 (X- ) 7.,
Xo