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.

220

Uniform convergence of an infinite series.

Bko 1II. ch. IV.

series in the interval from a to b converges, the series is uniformly
convergent for all values of x.
For then the number n of the place from which onwards the
remainders P,,, P+,, t... of the newly formed series are constantly
less than 6, assigns also a place such as is required for every x.
Examinple:
sinl x sin 3x sin 5 sin 7x
The series: -- - '^ + +
The ees 12 32 52 72
is uniformly convergent for all values of x, because:
1 f 1 1
12 32 52 72
is convergent (~ 47, foot-note p. 82).
For a series whose terms alternate in sign within an interval,
the following also is a sufficient criterion: The series is uniformly
convergent when for each number 6 a place n can be discovered in the
series, such that for every value of x the numerical values of the
terms from f,(x) onwards decrease and are constantly less than 6o
For, putting:
n,(x) - () fn (X) - (n) +;(X) t+n 2l(x) - f +(x) - () +
where the quantities f are all positive, Rn(x) is greater than 0 but
less than /f(x), because:
Rn (x) = [fn (x) -- f+l(x) -+ [n + 2(X) - fn+ (x)] 3 - + o
Rn (x) = fn(x) - [fn +l(X) -- fn + (X)] -- [n+3(X) -- fn+ 4 (X)] -
128. When an infinite series is uniformly convergent in the
neighbourhood of a point of its convergency, the infinite series expresses
a continuouts function at this point.
Denoting the sum of the first n - 1 terms of the series by 2(x),
we have:
F(x) = 2(x) + Rn (x).
Since the series is uniformly convergent in the neighbourhood of x,
a value n can be found such that abs R, (x) shall be less than - -6
for any value from x to x ~+ h, 6 being an arbitrarily small given
number. Hence, as:
abs [F(x - h/) - F(x)] = abs [2(x + _ h) —:(x) + R (x + h )- B (x)],
we have:
abs [F(x + h) - F(x)] < abs [z(x + h) - 2(x)] + - d.
But since the functions f are continuous and 22 is a sum of terms
finite in number, we can always choose a finite h so small that we
may have:
abs [2:(x + h) - 2:(x)] < - 6;