Colloquium publications.

DIVERGENT SERIES AND CONTINUED FRACTIONS. 81

simultaneously for a large number of consecutive values of m, the
best possible value for the function consistent with our information would evidently be obtained by carrying the computation
until the term of least absolute value is reached and then stopping.
Herein is probably the justification for the practice of the computer in so doing.
Equation (4) which gave a limit to the error in stopping with
the (m + 1)th term shows also that this limit grows smaller as x
diminishes. Since, furthermore, by increasing m sufficiently the
(m + 2)th term of (1) may be made small in comparison with the
(m + 1)th term, it is clear that on the whole, as x diminishes, we
must take a greater and greater number of terms to secure the best
approximation to the function. These two facts may be comprised
into a single statement by saying that the approximation given
by the series is of an asymptotic character. This will hold
whether the series is convergent or divergent.
This notion can be at once embodied in an equation. From (4)
we have
(lim f(x) - a- ax - -. - a
x-=O+
-lim ) 0 (m= 1,2,...).
x=04+ X
This equation is an exact equivalent of the two properties just
mentioned and is adopted by PoincarH * as the definition of asymptotic convergence. More explicitly stated, the series (1) is said
by him to represent a function f(x) asymptotically when equation
(5) holds for all values of m.
It will be noticed that this definition omits altogether the
assumptions concerning the nature of the function with which we
started in deriving the series. Not only has the requirement of
unlimited differentiability within an interval been omitted but the
existence of right-hand limits for the derivatives as x approaches
the origin is not even postulated. If the value a0 be assigned to
*Loc. cit.
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