i9o8.] IN INFINITE SERIES. 117 



This rearrangement of the terms of series (15) into the n col- 

 vimns of the table is permissible, inasmuch as throughout this inves- 

 tigation only absolute convergence is considered. 



Cauchy's ratio test shows that each one of the n partial series 

 composed of the terms in each of the n columns of the table is con- 

 vergent when 



(17) ;?v^-^^/^(;^-/^)»-^-- 



II. In like manner, if the algebraic functions defined by the 

 equations 



( 2 ) ay^ + by'' -\- ex = o 



(3) ay^x -\- by^ -\- c ^^ o 



are expanded into power series in x by Lagrange's theorem, and if 

 X is made unity in this power series, it is found that the resulting 

 infinite series are convergent, provided 



(18) -,-^,> 



12. If condition (18) is satisfied, equation (2) determines n — k 

 and equation (3) determines k roots of the three-term equation 



ay" -f~ ^y^ -\- c^=o. 

 Either condition (17) or condition (18) must be satisfied, unless 



(19) '^^-'^^ k\7i - k^-''' 



If condition (19) is satisfied, Raabe's test shows that the series 

 obtained from equations (i), (2), (3) are all convergent. 



13. The convergency conditions for equations (i), (2), (3) may 

 be written by following these directions : 



(a) To the left of the sign of inequality stands a fraction whose 

 numerator contains the coefficient of the middle term of the three- 

 term equation 



ay'* -|- by^ -|- c == o 

 and whose denominator contains the product of the coefficients of 

 the end terms, the exponent of each coefficient being the difiference 

 of the exponents in the other two terms taken in order from left 

 to right. 



