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Leonhard Euler
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This is the translation of Leonhard Eulers paper De Seriebus divergentibus written in Latin into English. Leonhard Euler defines and discusses divergent series. He is especially interested in the example $1!-2!+3!-text{etc.}$ and uses different methods to sum it. He finds a value of about $0.59...$.
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E661 in the Enestrom index. This was originally published as Variae considerationes circa series hypergeometricas (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma function. He looks at the relations between some infinite products and integrals. He takes the logarithm of these infinite products, and expands these using the Euler-Maclaurin summation formula. In section 14, Euler seems to be rederiving some of the results he already proved in the paper. However I do not see how these derivations are different. If any readers think they understand please I would appreciate it if you could email me. I am presently examining Eulers work on analytic number theory. The two main topics I want to understand are the analytic continuation of analytic functions and the connection to divergent series, and the asymptotic behavior of the Gamma function.
Let $mathscr{X}$ be the set of positive real sequences $x=(x_n)$ such that the series $sum_n x_n$ is divergent. For each $x in mathscr{X}$, let $mathcal{I}_x$ be the collection of all $Asubseteq mathbf{N}$ such that the subseries $sum_{n in A}x_n$ is convergent. Moreover, let $mathscr{A}$ be the set of sequences $x in mathscr{X}$ such that $lim_n x_n=0$ and $mathcal{I}_x eq mathcal{I}_y$ for all sequences $y=(y_n) in mathscr{X}$ with $liminf_n y_{n+1}/y_n>0$. We show that $mathscr{A}$ is comeager and that contains uncountably many sequences $x$ which generate pairwise nonisomorphic ideals $mathcal{I}_x$. This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.
A translation of Kummer`s paper On certain definite integrals and infinite series
Dirichlet proves the general convergence of Fourier series, after pointing out errors in an earlier attempt by Cauchy. We transcribed from Crelles Journal (1829) with numerous typographical corrections, and added a completed bibliography. Dirichlet prouve la convergence generale de la series de Fourier, apr`es avoir montre des erreurs dans un essai par Cauchy. Nous avons transcrit de Crelles journal (1829) avec de nombreuses corrections typographiques, et avons ajoute une bibliographie compl`ete.
This is an English translation of the Latin original De summa seriei ex numeris primis formatae ${1/3}-{1/5}+{1/7}+{1/11}-{1/13}-{1/17}+{1/19}+{1/23}-{1/29}+{1/31}-$ etc. ubi numeri primi formae $4n-1$ habent signum positivum formae autem $4n+1$ signum negativum (1775). E596 in the Enestrom index. Let $chi$ be the nontrivial character modulo 4. Euler wants to know what $sum_p chi(p)/p$ is, either an exact expression or an approximation. He looks for analogies to the harmonic series and the series of reciprocals of the primes. Another reason he is interested in this is that if this series has a finite value (which is does, the best approximation Euler gets is 0.3349816 in section 27) then there are infinitely many primes congruent to 1 mod 4 and infinitely many primes congruent to 3 mod 4. In section 15 Euler gives the Euler product for the L(chi,1). As a modern mathematical appendix appendix, I have written a proof following Davenport that the series $sum_p frac{chi(p)}{p}$ converges. This involves applications of summation by parts, and uses Chebyshevs estimate for the second Chebyshev function (summing the von Mangoldt function).
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