44 Formula for the Summation of Infinite Series, 



By comparing the several steps of this process with those 

 in the working of the first example, we are led to conclude that 

 the aggregate of two series, like this last and that of the first 

 example, will always be accurately expressed in a finite frac- 

 tion, provided the factors in the denominator be even in num- 

 ber, and that the difference of the two series will also be a 

 finite fraction, if the number of these factors be odd. In all 

 other cases the sum and difference will involve tt 2 . Thus the 

 difference between the two series 



1 _J_ _J_ - ir* 5 



l 2 .2.3 + 2 2 .3.4 + 3 2 .4.5 + * C ' ~ 12 ~ ~8 



and 



+ A a .2 t- „ , - a + &C. = — - 



1 .2.3* ' 2.3.4 2 3.4.5 s r 12 4 



is — ; so that actually subtracting and dividing by 2, we have 



1 1 1 1 



+ &c. = — 



] 2 . 2 . 3 2 T 2 2 . 3 . 4 2 J S* . 4 . 5 2 J 16 ' 



this series is therefore the square of the series 

 1 1 1 



T7273 + 2 . 3 . 4 + 3 . 4 . 5 + &C# 

 It thus appears that every series of the form 



1 1 



+ &c. 



l 2 .2.3...(m- 1)t» 2 ' 2 2 .3.4...;»(wi+ l) s 



is accurately summable when m is odd, but not when m is 



even. 



Similar remarks obviously apply to the series 



1 1 



1 2 .3.5... {m — 2)m* + 3*. 5 . 7 ... m(m -f 2) 2 + ' 



the sum of which is a determinable fraction only when the de- 

 nominator of each term consists of an odd number of factors. 

 In other cases the sum involves ir % . 



Although in the foregoing illustrations the several terms of 

 each series are all connected together by the sign plus, yet a 

 glance at the general investigation will serve to show that the 

 several deductions hold when the terms are alternately plus 

 and minus. 



