A Treatise on the Calculus of Finite Differences |
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Page 91
... convergent portion is sufficient . Applications . 3. The general expression for Eu , in ( 7 ) , Art . 2 , gives us at once the integral of any rational and entire function of x . Ex . 1. Thus making u = x , we have - – 2 Ex * = 0 + fa'd ...
... convergent portion is sufficient . Applications . 3. The general expression for Eu , in ( 7 ) , Art . 2 , gives us at once the integral of any rational and entire function of x . Ex . 1. Thus making u = x , we have - – 2 Ex * = 0 + fa'd ...
Page 99
... convergent if the other is but slowly so . 1 Ex . Transform the series - 12 1 + 1 13 14 - & c . into a $ ( 0 ) = ( 0 +12 ) ( − 1 ) , more convergent form . Here .. we have by ( 21 ) 1 1 - 12 + & c . 13 = 1 ( 1 2 1 2 + + 2.12.13 4.12 ...
... convergent if the other is but slowly so . 1 Ex . Transform the series - 12 1 + 1 13 14 - & c . into a $ ( 0 ) = ( 0 +12 ) ( − 1 ) , more convergent form . Here .. we have by ( 21 ) 1 1 - 12 + & c . 13 = 1 ( 1 2 1 2 + + 2.12.13 4.12 ...
Page 100
... convergent part we obtain an approximate value of the function on the left hand side of the identity ; and also that the closeness of the approximation depends on the smallness of the first of the terms in the rejected portion . From ...
... convergent part we obtain an approximate value of the function on the left hand side of the identity ; and also that the closeness of the approximation depends on the smallness of the first of the terms in the rejected portion . From ...
Page 104
... convergent character , and find an ap- proximate value of the sum of each when x = 5 , that is , cor- rect to 6 places of decimals . 6. If u + u ̧x + u ̧x2 + & c . = ƒ ( x ) , shew that U2 V x + U2 Vx + 1 + & c . = ƒ ( 1 ) + ƒ ' ( 1 ) ...
... convergent character , and find an ap- proximate value of the sum of each when x = 5 , that is , cor- rect to 6 places of decimals . 6. If u + u ̧x + u ̧x2 + & c . = ƒ ( x ) , shew that U2 V x + U2 Vx + 1 + & c . = ƒ ( 1 ) + ƒ ' ( 1 ) ...
Page 123
... convergent or 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 ...
... convergent or 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 ...
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Algebra applied approximate arbitrary constant Arithmetic assume B₁ Bernoulli's numbers C₁ C₂ Calculus of Finite Cambridge chapter Christ's College cloth College complete primitive convergent Crelle Crown 8vo curve deduce denoted determine difference-equation Differential Calculus differential coefficients differential equation divergent divergent series dy dx English equal Examples expression factor Fcap Finite Differences formula functional equation geometrical given equation gives Grunert Hence independent variable indirect integral integral function Lagrange's formula limits Mathematical method obtain operation ordinates P₁ P₂ particular periodical function problem Professor quantity rational and integral Schlömilch Schools Second Edition second member shew singular solution solve student substituting summation supposed symbol Taylor's Theorem theorem tion u₁ u₂ unity Ux+1 Ux+2 V₁ v₂ Vx+1 whence Их
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