(2) Merten' s theorem. If u„ converges absolutely to U and converges, but not absolutely, to V, then /7v„ converges to UV.
(3) Abel's theorem. If Zu„ converges to U and converges to V and is convergent, then Zw„ converges to UV.
(4) G. H. Hardy' s theorem. If
converges to U and
converges to V and
The product of two convergent alternating series is convergent, if w„—> o, this condition being both necessary and sufficient. This theorem is due to Pringsheim who has shown that a sufficient condition is that Zu„v„ should be convergent.This is not necessary, nor can there be any necessary and sufficient condition based on the product If = v„.= the product is convergent. If and v„ = the product is not convergent, although in each case = (n It may be noted that, if Eit„ and 'v,, are both convergent, the series cannot be strictly divergent, although it may oscillate, the range of oscillation being possibly infinite.
Series of Terms Containing a Variable. Uniform Con vergence.—If the terms of a series contain a variable x, real or complex, and if the series Zit,(x) is convergent for values of x in a certain interval, if x is real, or within a certain region, if x is complex, the sum of the series, which we may denote by S(x), determines a function of x within that interval (or region). If each term of the series is a continuous function of the variable, it does not follow that the function S(x) is continuous. For if we find the limit of S„(x), as n tends to infinity, for general values of x and then give x a particular value a, it is not at all certain that we shall obtain the same result as if we put x equal to a, before proceeding to the limit. Thus, if S„(x).-4o for all values of x, positive or negative; but S„(o) = and therefore the sum of the series, when x = o, is 1. Thus the function determined by the sum of the convergent series in which = -1-nx) is discontinuous at x = o. Discontinuity in a function defined by the sum of an infinite convergent series of continuous functions is always accompanied by a phenomenon known as non-uniform convergence. The term uniform convergence
is due to Weierstrass and the idea, though not unknown to Abel, was developed about the middle of the i9th century by him and by Sir G. G. Stokes, Seidel and Arndt. The fundamental notion of uniform convergence is this:—the formal condition that the series should be convergent is that, given any arbitrarily small positive number, E, it is possible to find an integer v, such that —S„(x) I < E, for all values of p if n-5. v. The least value of v, or the least that can be found in practice, depends not only on e but on x, and may be denoted by v(x). It may happen that an integer K can be found, depending only on E and not on x, such that for every value of x in the interval (or region) of con vergence K>v(x). In this case the series is said to converge uniformly to S(x) in the interval (or region). If however it hap pens that, N being any arbitrarily chosen number, however large, we can always find a value of x in the interval, such that v(x) > N, then the series is non-uniformly convergent. If we con sider = i/( +nx) and determine v such that IS„(x) I < e, v, we must have v>_.: — i)/x, x positive, and v>. i)/rx I , x negative. Thus in any interval which does not include zero, there is uniform convergence; for, if a is the lower limit of the interval if x is positive, or —b is the upper limit of the interval if x is negative, we can take v to be the next integer above — i)/a or (ci I) / b . But evidently we can take x sufficiently near to zero to require v> N, any assigned integer, however great. Hence the series is not uniformly convergent in any interval which includes zero.
One of the most important properties of a function defined by a convergent infinite series of functions of a variable is that it is continuous in any interval (or region) in which the series is uniformly convergent. If the series has a discontinuity at any point, it cannot be uniformly convergent throughout any interval containing that point. It does not however follow that non uniform convergence necessarily involves discontinuity. If s„ (x) = x/ (I -I-nx), the sum of the series for all values of x, zero included, is zero, but the series is not uniformly convergent in a range which includes zero.