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.

~ 44-45,

Properties of trigonometric series.

77

f'(X) == a + 2 a2 X + 3a3 x2 + * * * (n - 1) an-_ Xn-2 +..,
as was to be proved. The regressive differential quotient of the infinite
series of powers is a continuous function of x. For the progressive
differential quotient we obtain in the same way the same series.
45. Applying these Theorems to the trigonometric series
X 23 C5 X7 X2 n J+1
_, f E +- ' +(- - 1 )n + + * = Sill XX9 XI 6 2n
16 - ' P (- I + ' -- + COS X
we perceive that each of them converges absolutely for all finite values
of x, for we have for n = oo:
[2 2_+1 _ 2 n-l x2
Lim Lcv~l _ = Lim L +n
L. 2 T 2 I - 21 Fn r 2 1v 1I
Liral m2 1;- — 2 Lira I( —:. -.
12_ 12 _ - L(2 - )o
Accordingly the functions expressed by the series are continuous for
all finite values of x. Further we have
1) d sin x2 ', X2n
c x- - T- * * * (1- l)n *n e - i; e. cos x.
2n -1
d cos x x x3 x53 X
d icos _. 3 _.,., i. _. e. -- sin x.
dx - n 1 ( l) 1 _... D
Next it follows from the series that
2) cos (- x) = Cos X, sin (- x) = - sin x, cOS (0) - 1, sin (0)= 0.
Now, since equations 1) teach that all the derived functions even for
= cc remain finite and continuous, we may develope cos (x +- y)
according to Taylor's series in powers of y, and thus obtain:
d cosx y d2 cos x y3 d C cos x 4y' Cd cos x
cos(x +y) )=cos+ d- +y 2 d- - + 13 t- + -L -d -
Y2 Y3 L x + q D..
cosx-ysnx - c sin X; - c Cos x + -
eosy(l 2 + -* - i- +
i. e.
3) cos (x + y) == cos x cos n - sin /.
In like manner we find:
sin (x +- y) = sin x cos y -- cos x sin y.
Thus the theorem of addition, on which our previous calculations
were based, is proved independently of geometrical considerations.
From 3) we get, putting - y for y, in consequence of 2):
cos (x - y) == cos x cos y + sin x sin y.