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.

228

Integration of an infinite series.

Bk. III. ch. IV.

is a uniformly convergent series. Moreover even at the limits of the
convergency this series of powers is a continuous function, although
only semiconvergent (~ 44. IV); thus it always converges uniformly.
Second: The series derived by differentiating with respect to x:
f'(x) { a, + 2a2f(X)- + 3a3(f(x))2 +.. na(f(x))-1 + - v ),
as long as it converges, is continuous and expresses the derived of
the given series; but it certainly converges within the interval of the
original series.
Third: The integral of the given series, taken between two
values xO and x1 in the convergency, is formed by the uniformly
convergent series:
X1 X1 xi
ao + a, Jf(x)dx + a2J {f(x)}2dx+..+a {f(x)}I dx + etc..
xo Xo Xo
If this series converge also at the limits of the convergency and
remain continuous there, it expresses the integral up to and including
the limits.
x
132. Expression of the function sin-1 x- - -dx by a series.
If x2 < 1 we have the expansion:,1 ' 1 2 +1.3 4 1.3.5 6 +... 1.3....(2 —l)x..
(1 —x2)a = — I +2- + 2- 4 X4 - 2.4.6 X 2. 4.6... 2n
hence:
i x3 1.3 x5 1.3.5 X7 1.3... (2n - 1) x2n-+1
sin-Xy ----x — 5 - 24 -'-{ -.2n2....2n n+ etc..
s2 3 t2.4 _5 2.4.6 7 2,4...2 n 2n +1
This series continues to converge even for x2 - 1, although the above
binomial series is no longer convergent. For, the terms of the series:
1 1 1 1.3 1 1.3.5 1 1.3.5.7 1
1 2 3 2.4 5 2.4.6 7 2.4.6.8 9
are smaller than the corresponding terms of the series:
+ + 1.3 + 1.3.5 + + et
2 2.4 2.4.6 2.4.6.8
But this series converges and its value is 2. For:
1 — x X2.3 3
2 2 4 2.4 6
converges even for the value x =-1. Therefore:
1 I I 13 1.3.5 1
sin-I(1)- 2 + - - ' + 2 4. - etc..
e dinit1e_ 2 3g l 2-.4 5 2.4.6 7
The definite integral f/d 2as already indicated in 107, has the
0