A Treatise on the Calculus of Finite Differences |
From inside the book
Results 1-5 of 15
Page 65
... divergent accord- ing as the sum of its first n terms approaches or does not approach to a finite limit when n is indefinitely increased . This definition leads us to distinguish between the con- vergency of a series and the convergency ...
... divergent accord- ing as the sum of its first n terms approaches or does not approach to a finite limit when n is indefinitely increased . This definition leads us to distinguish between the con- vergency of a series and the convergency ...
Page 66
... x varies continuously from a to ∞ , then the series $ ( a ) + $ ( a + 1 ) + $ ( a + 2 ) + & c . ad inf ...... ( 3 ) . ... .00 will be convergent or divergent according as [ * 66 CONVERGENCY AND DIVERGENCY OF SERIES . [ CH . V.
... x varies continuously from a to ∞ , then the series $ ( a ) + $ ( a + 1 ) + $ ( a + 2 ) + & c . ad inf ...... ( 3 ) . ... .00 will be convergent or divergent according as [ * 66 CONVERGENCY AND DIVERGENCY OF SERIES . [ CH . V.
Page 67
George Boole .00 will be convergent or divergent according as [ * Þ ( x ) dæ is finite or infinite . a For since ( x ) diminishes from xa to x = a + 1 , and again from x = a + 1 to x = a + 2 , & c . , we have ra + 1 fa α ra + 2 4 $ ( x ) ...
George Boole .00 will be convergent or divergent according as [ * Þ ( x ) dæ is finite or infinite . a For since ( x ) diminishes from xa to x = a + 1 , and again from x = a + 1 to x = a + 2 , & c . , we have ra + 1 fa α ra + 2 4 $ ( x ) ...
Page 68
... divergent if h х 1 . The general term is h " , the value of x in the first term being 0 , so that the test of convergency is simply whether h * dx is infinite . Now hdx = hx 1 log h - · If h > 1 this expression becomes infinite with x ...
... divergent if h х 1 . The general term is h " , the value of x in the first term being 0 , so that the test of convergency is simply whether h * dx is infinite . Now hdx = hx 1 log h - · If h > 1 this expression becomes infinite with x ...
Page 69
... divergent if m1 . 5. Perhaps there is no other mode so satisfactory for establishing the convergency or divergency of a series as the direct application of Cauchy's proposition , when the inte- gration which it involves is possible ...
... divergent if m1 . 5. Perhaps there is no other mode so satisfactory for establishing the convergency or divergency of a series as the direct application of Cauchy's proposition , when the inte- gration which it involves is possible ...
Other editions - View all
Common terms and phrases
A₁ A³u algebraic applying arbitrary constant ascending powers assume ax+b B₁ c₂ Calculus of Finite Cambridge CHARLES KINGSLEY cloth CO.'S PUBLICATIONS complete primitive CONIC SECTIONS convergent Crown 8vo curve deduce denoted determine Differential Calculus differential equation divergent English Notes equation of differences expression Fcap Fellow of St Finite Differences functional equation GEORGE BOOLE given equation Hence independent variable indirect integral John's College limit linear equation M.A. Second Edition MACMILLAN & CO.'S method mixed differences operation ordinates P. G. TAIT particular value periodical function rational and integral reduced result Schools second member Sermons shew singular solution substituting successive values supposed symbol theorem tion TODHUNTER Treatise Trinity College u₁ u₂ unity University of Cambridge Ux+1 Ux+2 v₂ Vx+1 Vx+2 whence Их
Popular passages
Page 2 - FROST.— The First Three Sections of Newton's Principia. With Notes and Problems in illustration of the subject. By PERCIVAL FROST, MA late Fellow of St. John's College, Cambridge, and Mathematical Lecturer of Jesus College. Crown 8vo. cloth, 10s.
Page 6 - THE SEVEN KINGS OF ROME. An Easy Narrative, abridged from the First Book of Livy by the omission of Difficult Passages; being a First Latin Reading Book, with Grammatical Notes and Vocabulary.
Page 2 - Scouring of the White Horse. Or, the Long Vacation Ramble of a London Clerk. By the Author of
Page 5 - The fitness of Holy Scripture for Unfolding the Spiritual Life of Man : Christ the Desire of all Nations ; or, the Unconscious Prophecies of Heathendom.
Page 15 - JW COLENSO, DD Bishop of Natal. The Colony of Natal. A Journal of Visitation. With a Map and Illustrations.
Page 1 - Prelector of St. John's College, Cambridge. AN ELEMENTARY TREATISE ON MECHANICS. For the Use of the Junior Classes at the University and the Higher Classes in Schools.
Page 5 - SPECTATOR. 6. Alexandria and Her Schools. Being Four Lectures delivered at the Philosophical Institution, Edinburgh. With a Preface. Crown Svo. cloth, 5*. " A series of brilliant biographical and literary sketches, interspersed with comments of the closest modern, or rather universal application
Page 4 - THRUPP.— Antient Jerusalem : a New Investigation into the History, Topography, and Plan of the City, Environs, and Temple.
Page 5 - Village Sermons. By a NORTHAMPTONSHIRE RECTOR. With a Preface on the Inspiration of Holy Scripture. Crown 8vo. 6s. Vittoria Colonna. — Life and Poems. By MRS. HENRY ROSCOE. Crown 8vo. 9s. Volunteer's Scrap Book. By the Author of "The Cambridge Scrap Book.