Infinite Series

Original Titles
English Titles

19On transcendental progressions, that is, those whose general terms cannot be given algebraically
20The summation of an innumerable progression
25A general method for summing series
41On the sums of series of reciprocals
43On harmonic progressions
46Universal methods of series
47Finding the sum of any series from a given general term
55Universal method for summation of series, further developed
61On sums of series of reciprocals from powers of natural numbers from another discussion, in which the sums are derived principally from another source
63Demonstration of the sum of the series 1 + 1/4 + 1/9 + 1/16
71A dissertation on continued fractions
72Various observations about infinite series
74On various methods for expressing the quadrature of a circle with verging numbers
122On products created from infinite factors
123Observations on continued fractions
125Consideration of a progression suitable for finding the quadrature of a circle
128An easy method for computing the natural and artificial sines and tangents of angles
130Considerations on certain series
189On the determination of series, or a new method for finding the general terms of series
190Consideration of certain series which are gifted with particular properties
246A contribution to the calculations of sines
247On divergent series
275Annotations to a certain passage of Descartes for finding the quadrature of the circle
280On progressions of arcs of circles, of which the accompanying tangents proceed by a certain law
281A specimen of a singular algorithm
326Analytical observations
352Remarks on a beautiful relation between direct as well as reciprocal power series
393On the sum of series involving the Bernoulli numbers
432Analytical exercises
447The summation of the progressions
sin(φλ) + sin(2φλ) + sin(3φλ) + ... + sin(nφλ);
cos(φλ) + cos(2φλ) + cos(3φλ) + ... + cos(nφλ).
453Eminent properties of series within which the general term is contained as x = (1/2)(a+b/√k)(p+qk)n + (1/2)(a-b/√k)(p-qk)n
465A demonstration of a theorem of Newton on the expansion of the powers of a binomial by cases, in which the exponents are not integral numbers
477Meditations about a singular type of series
489On unravelling exponential formulas
507On the infinity of infinities of orders of the infinitely large and infinitely small
522On the formation of continuous fractions
550On series in which the product of two consecutive terms make a given progression
551Various methods for inquiring into the innate characters of series
553Analytical observations
555An examination of the use of interpolating methods in the doctrine of series
561Various observations about angles proceeding in geometric progression
562On how sines and cosines of multiplied angles may be expressed by products
565On highly transcendental quantities, which may not be expressed in any way by integral formulas
575De mirabilibus proprietatibus unciarum, quae in evolutione binomii ad potestatem quamcunqua evecti occurrunt
583De numero memorabili in summatione progressionis harmonicae naturalis occurrente
584De insignibus proprietatibus unciarum binomii ad uncias quorumvis polynomiorum extensis
592On the resolution of transcendental fractions into infinitely many simple fractions
593On the transformation of series into continued fractions, where at once this not mediocre theory is enlarged
597A new and most easy method for summing series of reciprocals of powers
613Dilucidationes in capita postrema calculi mei differentalis de functionibus inexplicabilibus
616On the transformation of the divergent series 1 - mx + m(m+n)x2 - m(m+n)(m+2n)x3 + etc. into a continued fraction
617On the summation of series, in which the signs of the terms alternate
636On the multiplication of angles which are to be obtained by factors
637A new demonstration, with respect to which prevails the expansion of binomial powers by Newton even by fractional exponents
642On a singular rule for differentiating and integrating, which occurs in the sums of series
652On the general term of hypergeometric series
655General observations about series, of which the terms arising for the sines or cosines of multiplied angles come forth
661Several considerations about hypergeometric series
663Plenior expositio serierum illarum memoragilium, quae ex unciis potestatum binomii formantur
664Analytical exercises
684On the roots of the infinite equation 0 = 1 - (xx)/(n(n+1)) + (x4)/(n(n+1)(n+2)(n+3)) - (x6)/(n.....(n+5)) + etc.
685An analytical exercise, where in particular a most general summation of series is given
686Elucidations about the formula, in which the sines and cosines of angles are to be multiplied, where at once large difficulties are diluted
703An easy method for finding series proceeding by the multiplication of the sines and cosines of angles, of which the use in the universal theory of astronomy is very great
704Disquisitio ulterior super seriebus secundum multipla cuiusdam anguli progredientibus
705Investigatio quarundam serierum, quae ad rationem peripheriae circuli ad diametrum vero proxime definiendam maxime sunt accommodatae
706On a new type of rational and highly convergent series, by which the ratio of the periphery to the diameter is able to be expressed
709On the expansion of the power of any polynomial 1 + x + x2 + x3 + x4 + etc.
710Example of the transformation of singular series
722Analytical disquisitions on the expansion of the trinomial power (1+x+xx)n
726A demonstration of a notable theorem of numbers a twelfth part of binomial powers
736On the summation of series contained in the form a/1 + a2/4 + a3/9 + a4/16 + a5/25 + a6/36 + etc.
742Observations about continued fractions contained in the form S = n/(1+(n+1)/(2+(n+2)/(3+(n+3)/(4+etc.))))
743De serie maxime memorabili, qua potestas binomialis quaecunque exprimi potest
745On the continued fractions of Wallis
746A method for gathering the sums of infinite series by investigating differential formulas
747On remarkable series, by which the sines and cosines of multiplied angles may be expressed
750A commentary on the continued fraction by which the illustrious La Grange has expressed the binomial powers
768De unciis potestatum binomii earumque interpolatione
809Series maxime idoneae pro circuli quadratura proxime invenienda
810Enodatio insignis cuiusdam paradoxi circa multiplicationem angulorum observati
819Continuation of some fragments taken from the Mathematics day book