Series

A power series converges absolutely and uniformly throughout the interior of its circle of convergence. It can be integrated or differentiated term by at all points of the interior of the circle. Two power series in the same variable cannot represent the same function in any region without being identical.

An important theorem on power series is Abel's theorem. If Ea„x" converges to f(x) for lx I < I, and if Ea„ converges to a sum S, then limf(x)x-->i — =S.

But the existence of limf(x) x-- 1— does not necessarily imply the convergence of It was shown by Tauber that the condition is sufficient to secure the convergence of if limf(x)x-- i — exists. More recently J. E. Littlewood has shown that the condition

Non-convergent Series. Summability. Asymptotic Series. —A series that is not convergent cannot have a sum in the ordinary sense. Nevertheless it is possible to modify the defini tion of the sum of a series so that a sum can be assigned to a non convergent series and the sums of such series can be used in ap plications as if they were the sum of ordinary convergent series. Euler used such sums to obtain accurate results. Their use by writers whose mathematical insight was less acute than Euler's frequently led to grave errors and great mathematicians of the early i9th century, such as Abel and Cauchy, deliberately ex cluded from analysis the use of any such series and refused to accept as sound any demonstration involving their use. Towards the end of the i9th century interest in such series was revived and their treatment put on a sound footing. The subject of such series has received much attention in this century and many important contributions have been made to it by English mathematicians. Only the briefest sketch of it can be given here. The idea of the sum of a non-convergent series is based on the following principle. If certain limiting operations connected with a con

vergent series result in a finite limit which is the sum of the series, it may happen that the same operations applied in connection with a non-convergent series result in a finite limit. The non convergent series is then said to be summable and the limit thus found is said to be its sum. For example, if the series • • gent with a sum S, then S is the limit of f(x) as x approaches from below. But f(x) may tend to a limit in this way without 2.a„ being convergent and we may regard this limit, if it exists, as the sum of the series when it is not convergent. Thus the series I — x • converges to if x < I, and we may regard the series I—i+I—I+ • • • as having a sum, which is 1. A classical method of summing a non-convergent series, due to E. Borel, is the following. The integral has a definite value. In this case the series is said to be summable and the resulting sum to be "Borers sum." If we apply this method to the series I — • • • we obtain as the sum (1 which is the actual sum when the series is convergent. Another classical method is that of E. Cesar°. If has a limit, • • • +an)ln has the same limit, but may have a limit, although oscillates. Hence, although the sequence • • • may not have a limit, it may happen that • • -1-sn)/n has a limit. If this happens, the series is said to be summable (CI). The Cauchy product of any two convergent series is summable (CI). If this mean has not a limit we may repeat the operation, and so on any number of times. Actually Cesar° uses as his rth mean the quotient of • • • by i +A,+ • • where A, is the coefficient of xr in the expansion of (I by the Binomial Formula (q.v.). If this rth mean gives a limit the series is said to be summable (Cr). A series, summable (C, r) is also summable (C, r+ I) and so on. It is not true that an arbitrary series can be summed by any of Cesaro's means or by any known method. A striking and im portant theorem connected with Cesaro's means, due to G. H. Hardy, is that a series which is summable (CI) is convergent, if its terms are such that na„