Method of Progressions, 201 



constant quantity introduced in the above piogression for the 

 purpose of rendering it capable of being compared with progres- 

 sions into which constant quantities enter. 

 As an example I shall take the progression 



~;Ti^(^ + (^ + 2'') + (2"+3")+ (m-l)"+m") 



which, I have shown in my First Letter, will express the area of a 

 curve of which the ordinate is ax and the al)scissa x. 



The last term of this progression is — : x (^w— 1 -i-fn \ 



And to make the last difference of the assumed progression con- 

 tain the same powers of m, when reduced, it must be 



(md) —{^[m — \]d) —d xlm — [m — 1] ). 



Making 



— :xnm — 1] +m)=d x(m — fm — 1] ), we 



2m 



n + 1 



r n " , n 



[m — 1 J + m 



find d = : X (—^, — —^ V But the sum of the 



2ni TO — [to— 1] 



progression Will be [nid) = — - — x( p j. 



TO — [to— 1]" 



If the numerator and denominator of the latter part of this 

 expression be expanded by means of the binomial theorem, we 

 obtain the sum of the progression under the form 



— 1 —2 n — 1 "— 2_. — 3 



„ , I B» * —!*//» '/t ri 7rt -j- &c. 



X 



— I , n — 1 —2 n— 1 n — "i — a 



H+ I J 1 —,ii)i -\-n . — _ m n . to +&c. k 



r 3 3 4 1 



It will readily be seen that the sum of the progression can 



n+ 1 



never be accurately represented by — except when 72 = 1 ; but 



it is evidently nearer to the true sum in proportion as the value 

 of m is increased. 



Let AC be a straight line, and make AB, ab, afb', a"h" , &c. 

 each perpendicular to it ; then, if AB, aZ», ab' , &c. represent 



• . . «,"■*"' 

 the successive differences between — and the true sum of tlie 



71+ I 



progression corresponding to certain increased values of m, and 

 if a line be drawn tinongh the j)oints B, /', b' y &;c. it will con- 

 tinually appioath the line AC, as a curve does to its asvmptote ; 

 Vol. o7. No. 275. March lb21. C c ' but 



