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for a given x, but the error in this case cannot be made as small as we wish. However,
we are able to minimise the error, for a given x, by retaining a precise number of terms
in the series–one term more or one less will increase the error. The number of terms
retained will depend on the value of x at which f (x) is to be estimated. This important
property can be seen in the case of a (divergent) series which has alternating signs—a
quite common occurrence—via a general argument.
Consider the identity

where N is finite; is the remainder. Suppose that and with
(and, correspondingly, a reversal of all the signs if this
describes the alternating-sign property of the series. Let us write

then

But the remainders are of opposite sign, so they always add (not cancel, approximately),
which we may express as

similarly

Hence the magnitude of the remainder—the error in using the series—is less than the
magnitude of the last term retained and also less than that of the first term omitted. It is
important to observe that, provided N remains finite, it is immaterial to this argument
whether the series is convergent or divergent. Thus, for a given x, we stop the series
at the term with the smallest value of (which, if the series is convergent, arises at
infinity and is zero); the sum of the terms selected will then provide the best estimate
for the function value. Let us investigate how this idea can be implemented in a classical
example.

E1.5 The exponential integral
A problem which exhibits the behaviour that we have just described, and for which
the calculations are particularly straightforward, is the exponential integral:

We are interested, here, in evaluating Ei(x) for large x (and we observe that
as see Q1.13); of course, we cannot perform the integration, but we can