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MR. E. W. BARNES ON INTEGRAL FUNCTIONS. 
Part II. 
The Theory of Divergent Series. 
§ 23. The development of the theory of divergent series is an interesting instance 
of the progress of mathematical thought. The beginning was purely arithmetic : to 
find some approximation to the value of n !, where n is a very large integer.* In the 
result it appeared that the value of log n ! could be more and more nearly calculated by 
adding on successive terms of a series proceeding by powers of The error is of the 
order of magnitude of the term of the series next after the one at which we stop. 
And, most important fact of all, the series is divergent. 
If n ! he replaced by V (n + 1), a similar result can be obtained, which holds for all 
real positive values of n. 
Finally, there comes the enquiry as to what meaning, if any, can he attached to the 
equality in the case in which n is any complex quantity. 
Other approximations undergo the same process of development, so that it becomes 
necessary to try and construct a formal theory. 
What we may call the arithmetic theory has been given by Poincare,! for the 
case in which all the quantities involved are real :—a restriction which the author 
subsequently assumes to be unnecessary. 
For the more extended case, when z is any complex quantity, we may say that the 
• CC (t 
divergent series a 0 + „ +•••+! + ••• of which the sum of the first (n + 1) 
Z Z 
terms in S„, will, when \z\ is very large, be an asymptotic expansion for a function 
J ( 2 ) if the expression \z n (J — S„) | tends to zero, as 2 tends to infinity. 
Thus, if 2 be sufficiently large, 1 2 " (J — S„) | < e where e is very small. 
The error J — S„ = e/z' 1 committed in taking for the function J the first n + 1 
terms of the series has a modulus which is infinitely smaller than the modulus of the 
error J — S„_ 1 = a u fi- e/z n obtained by taking only the first n terms, for | a n | is in 
general finite, and | e \ is very small. 
In view of subsequent results, it proves necessary to define the equality of the 
function and divergent series for values of 2 which lie along some definite line tending 
to infinity. We do not then assume that the expansion is possible all round the 
point 2 = 00 . 
It will be sufficient to recapitulate the results which Poincare obtains. 
We may multiply two asymptotic series together by the same rules as we should 
apply to absolutely convergent series. 
* Stirling, 1 Methodus Differentials ’ (1730). 
t ‘Acta Mathematical 8, pp. 295-344; ‘ Mecanique Celeste,’ vol. 2, pp. 12-14. 
