Series

If is real and one-signed, then log is one-signed and so also is log(' —u„). Since it follows that Elog(i -+- and Elog(i —u„) are each convergent or divergent according as is convergent or divergent. Hence if is real and positive II(i and MI are each con vergent, if /u.„ is convergent. If fat„ is divergent MI +2?„) diverges to 00 and MI —u„) diverges to zero.

If is either real or complex ilog(i± u„)1

If I'u„ is not absolutely convergent, we use the fact that if < I, log(r = where A„

If contains a variable x and is written as u„(x), then a convergent infinite product HI' defines a function of x. For such products there is a theory of uniform convergence similar to that developed in connection with infinite series.

Corresponding to Ta.nnery's theorem we have the theorem that, if F(n) =11114 - , and satisfies the conditions of Tannery's theorem, then F(n) tends to a limit, which is the value of the convergent infinite product We can by this theorem deduce from the identity, 0 Y7 sine = (2n+ I) sin 11 I I sin' I „, 2/Z+ I discovered by Wallis.

Weierstrass has shown that, if • • • is an infinite se quence, in which never decreases and tends to infinity with n, and is such that I is divergent, we can find nomials such that the infinite product is absolutely convergent for all values of z. Here we do not say that the series diverges for the values z= • • because, if the factor (I —z/a„) is omitted, the remaining infinite product is convergent when z =a„. We say that z is a zero of the function determined by the infinite product. One simple

example is provided by taking the sequence to be — I, - 2, — 3, • • • . The simplest factor that can be associated with (I +z/n) is so that the infinite product MI is con vergent. Since I +-1-1- • • • —logn tends to Euler's constant C, it follows that I) (z+ 2) • • • (z+n)/n! tends to a limit as n tends to infinity. The reciprocal of the function of z de fined by this limit has been denoted by Gauss by II(z). It is also F(z+ I), where F(z) is the Gamma-function. Further details of functions defined by infinite products belong to the theory of functions. As with infinite series, so with infinite products, we may have doubly and multiply-infinite double products, and the treatment of such products may be made to depend on series. The function defined by the doubly-infinite product where 0= and co, being any two numbers whose ratio is complex, and m and n have any positive or negative integral values, simultaneous zero values alone being excluded, is Weierstrass's Sigma-function, fundamental in the theory elliptic functions.

BIBLIOGRAPHY.-See

the article, "Algebraische Analysis," by A. Pringsheim and G. Faber, in the Encyklopddie der mathematischen Wissenschaften (Leipzig, 1898, etc., vol. ii., ch. i.). The elementary treatment of series may be found in W. P. Milne, Higher Algebra (1913) ; F. Bowman, Algebra (1925-27) ; and in great detail in G. Chrystal, Introduction to Algebra (1898, 4th ed. 192o). More advanced treatment can be found in G. H. Hardy, A Course of Pure Mathematics (1908, 4th ed. 1925) ; E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (1902, 4th ed. 1927) ; T. J. I'A. Bromwich, An introduction to the Theory of Infinite Series (1908, 2nd ed. 1926) ; E. W. Hobson, Theory of Functions of a Real Variable (1907, 3rd ed. 1927 etc.). Except that the 2nd ed. of Bromwich omits the question of summability and Hobson is re stricted to real variables, these last two works are practically exhaus tive. A very complete treatment is given in K. Knopp, Theorie and Anwendung der unendlichen Reihen (1922, Eng. Trans., 1928). For particular portions of the theory, refer to E. Borel, Lecons sur les series divergentes (19oI) and to others of the series of monographs on the theory of functions edited by Borel. Numerous important con tributions to the theory will be found in the publications of recent years of the London Mathematical Society and the Cambridge Philosophical Society. A paper by G. H. Hardy in the Proceedings of the C. P. S., vol. xiv. (May 1918), entitled "Sir George Stokes and the Concept of Uniform Convergence," will be found of particular interest. (A. E. J.)