122 Mr. 0. Heaviside. [.Time 15, 



series of this kind have the terms alternately positive and negative. 

 This seems to give a clue to the numerical meaning. The terms get 

 bigger and bigger, but the alternation of sign prevents the assump- 

 tion of an infinite value, either positive or negative. It is possible to 

 imagine a, finite quantity split up into parts alternately positive and 

 negative, and of successively increasing magnitude (after a certain 

 point, for example) . It is a bad arrangement of parts, certainly, but 

 understandable roughly by the initial convergence. So the use of 

 alternating divergent series may be justified by numerical convenience 

 in an approximate calculation of the value of the function. 



But, by the same reasoning, a direct divergent series, with all terms 

 of one sign, is of infinite value, and therefore out of court. It cannot 

 have a finite value, and cannot be the solution of a physical problem 

 involving finite values. This seems to be what Boole meant in his 

 remark on p. 475 of his ' Differential Equations ' (3rd edition) : " It 

 is known that in the employment of divergent series an important 

 distinction exists between the cases in which the terms of the series 

 are ultimately all positive, and alternately positive and negative. In 

 the latter case we are, according to a known law, permitted to employ 

 that portion of the series which is convergent for the calculation of 

 the entire value." He proceeded to exemplify this by Petzval's in- 

 tegrals. The argument is equivalent to this. Change the sign of x 

 in the Series C, equation (3) above. Let the result be C'. Then we 

 must use the Series C' when x is positive, and C when x is negative. 

 This amounts to excluding the direct divergent series altogether, and 

 using only the alternating. That is, we have one solution, not two. 

 Professor Boole did not say what the " known law " was. His above 

 authoritative rejection of direct divergent series led me away from the 

 truth for many years. The plausibility of the argument is evident, as 

 evident as that the value of a direct divergent series is infinity. 



Divergent Series as Differentiating Operators. 



45. (d). Later on, divergent series presented themselves in an 

 entirely different manner. In the solution of physical problems by 

 means of differentiating or analytical operators, the operators them- 

 selves may be either convergent, or alternatingly divergent, or directly 

 divergent. That is, they are so when regarded algebraically, with a 

 differentiator regarded as a quantity. When the operations indicated 

 by the operator are carried out upon a function of the variable, the 

 solution of the problem arises, and in a convergent form. Here, then, 

 we have the secret of the direct divergent series at last. It is nume- 

 rically meaningless, when considered algebraically, with a quantity 

 and its powers involved. But analytically considered, the question 

 of divergency does not arise. The proper use of divergent series is as 



