 About 1,420 results  books.google.com and that Cayley was in 1893 still ignorant of the interesting and valuable
investigations made by Darbonx,* Du Bois ... The only points of nonuniform
convergence known to him were those for which the sum of the series is
discontinuous ; and as ... 127—(l and from this he derives his criterion, viz., that
the series iU(tll') must have “ a sum indefinitely large for a: ... T Du Bois Raymond
, Abh. d. illii'nch. 

 books.google.com By putting ¢(n) : u,,<p(n), where q1(n) is ositive, Du Bois Reymond's general
criteria of the second ind are obtained : if um ... who showed how from any
convergent (divergent series a new series can be obtained which converges (
diverges; ... 

 books.google.com then the given series converges whenever limit ip(ri) remains finite, and limit \n
does not become zero ; the series diverges whenever ... Criteria for continually
weaker degrees of convergence or divergence are obtained by substituting for ip{
ri) a succession of ... By putting ip{ri) = un<p(n), where <p(n) is positive, Du Bois
Reymond's general criteria of the second kind are obtained : if lim  <p(n)  ^±J_ 9
(n. 

 books.google.com As late as 1844 De Morgan began a paper on “divergent series” in this style: “I
believe it will be generally admitted that the ... criteria never fail; but Du Bois
Reymond and Pringsheim have each discovered series demonstrably
convergent in ... 

 books.google.com then the given series converges whenever limit if:(n) remains finite, ana limit in
does not become zero ; the series diverges ... Criteria for continually weaker
degrees of convergence or divergence are obtained by substituting for V'(w) a ...
By putting ip(n) = u,<p(n), where <p(n) is positive, Du Bois Reymond's general
criteria of the second kind are obtained : if lim j <p(n)  n = oo then 2 u, is
convergent. 

 books.google.com ... criterion for establishing the convergence of a Fourier series. In spite of some
sharp criticisms of du Bois–Reymond's claim to have established a necessary
condition for convergence, Weierstrass recognised that du Bois– Reymond's
works ... 

 books.google.com First in time in the evolution of more delicate criteria of convergence and
divergence come the researches of Josef Lud wig ... It was the opinion of Bonnet
that the logarithmic criteria never fail ; but Du BoisRey mond and Pringsheim
have each discovered series demonstrably convergent in ... The study of general
criteria was continued by U. Dini of Pisa, Paul Du BoisReymond, G. Kohn of
Minden, and ... 

 books.google.com where a > 0, and divergent if Similarly the integral / f(x) dx is convergent if f< (1/ar)
{/ (1/*) . .. 41 (I/*)}1 {h (l/*)}~1", ... The first general statement of the ' logarithmic
criteria', so far as series are concerned, appears to have been made by De
Morgan, 1, 325. The essentials ... We should mention also certain theorems of a
negative character, analogous to du BoisReymond's theorem of § 2.1. Given any
... 

 books.google.com For the series 2 1/<£ (n) is either convergent or divergent, and in either case the
criterion is sufficient. This criterion, which may ... That this latter restriction is un
necessary was shewn by Dini and by Du BoisReymond. Companion criteria of ... 

 books.google.com For examples of series and integrals which converge or diverge so slowly as not
to answer to any of the logarithmic criteria see Du BoisReymond, Pringsheim,
Borel (I.e. supra), and Blumenthal, Principes de la theorie des fonctions entires ... 

 