A less artificial looking series is the series I sin nO, which is convergent for all real values of 0. If o
Tests for uniform convergence are not numerous nor are they always very simple. One very simple test, which covers most applications, is known as the Weierstrass M-test. It is this. If a number U can be found such that lu(x) I < for all values of x in an interval (or region) and I U is convergent, then is uniformly convergent throughout the interval.
Thus the series .Exn cos{4)(x) } , where is any real function of x, is uniformly convergent in any interval a to b, where I a l and I bl are each less than unity.
It has been remarked that the series ZasinnO, where decreases steadily to zero, is convergent for all real values of 0. Such a series is uniformly convergent in any range that does not include zero or a multiple of 27r. That it should be uniformly convergent in any range whatever it is necessary and sufficient that should tend to zero. This is practically the only type of series for which a condition for uniform convergence, necessary as well as sufficient, exists.
If
converges uniformly to S(x) throughout the interval a to b, then fS(x)dx = f u(x)dx, if both the upper and lower limits of integration are not outside the interval. It is not ab solutely necessary that uniform convergence should exist for term-by-term integration, as it is called, to be valid, but no better completely general conditions are known. One case may be mentioned where term-by-term integration is valid over a range which contains a point of non-uniform convergence and that is if
I
For term-by-term differentiation we have the theorem that = S(x) at all points of an interval throughout which dx dx the series on the left-hand side is uniformly convergent. But this condition is not necessary.
It should be noted that even when the series is uniformly convergent throughout any interval, however large its upper limit, term-by-term integration with infinity as the upper limit is not always possible. The series for viz.
series of integrated terms not being convergent otherwise.
Absolute convergence and uniform convergence are quite distinct. Neither implies the other. The series I( where x is real, is uniformly convergent in any interval. For
IS,p(x) I < < On+ I) and therefore < E, if n+ I > a condition independent of x. Yet the series is not absolutely convergent for any value of x. On the other hand the series /II(' +nx) i/(i+n+ ix)} is absolutely convergent for all values of x, but is not uniformly convergent throughout any interval which includes x = o.
The following important theorem, known as Tannery's theo rem, is based upon an idea similar to that of uniform convergence. Let u2(n), be a sequence of numbers each involving n.
Let S(n) where the series is infinite or stops at a term where p depends on n and becomes infinite with n. Let tend to as n tends to infinity, and let it be possible to find independent of n, such that I < for all positive values of n and the series EU, is convergent. Then S(n) tends to a limit, as n tends to infinity, which is the sum of the series Zit,. By means of this theorem we can prove that the exponential series IznAn!) is the limit of + n as n a positive integer, tends to infinity. We can also deduce the expansions of cos 0 and sine as power series in 0 from the identities or the reciprocal of the limit of I if this limit exists. If neither of these limits exists, the radius is the reciprocal of the upper limit of that is, the greatest number G such that an infinite set of integers (not consecutive) can be found such that for every such integer IG I I inn I < e, where E is an ar bitrarily assigned small positive number. For instance, if = sin ne, G is evidently unity, though neither of the other limits exists. If x is restricted to be real, then x must lie on that diameter of the circle which is the axis of real numbers. The circle is called the circle of convergence. The radius of the circle may in some cases be indefinitely great, as in the case of the exponential series and the sine and cosine series, which are con vergent in the interior of any circle, however great its radius, or the circle may shrink to a point as with the series i !-F 2 which is convergent for no value of x except zero. For points on the circle itself, the series may or may not be convergent. For n(n I) example, if we take the binomial series I +nz+ , , where n is not a positive integer, the circle of convergence is the circle I al = I. If n is positive the series converges absolutely for every point on the circle. If n is negative but n+ I is positive, the series converges, but not absolutely, at every point of the circle except z = I. If n+ I is not positive, the series does not converge at any point of the circle.