Our definitions of the sum of an infinite series, and the value of an infinite integral, whether of the first or the second kind, apply to series of terms or integrals of functions whose values may be either positive or negative. But the special tests for convergence or divergence which we have established in this chapter, and the examples by which we have illustrated them, have had reference almost entirely to the case in which all these values are positive. Of course the case in which they are all negative is not essentially different, as it can be reduced to the former by changing $$u_{n}$$ into $$-u_{n}$$ or $$\phi(x)$$ into $$-\phi(x)$$.

In the case of a series it has always been explicitly or tacitly assumed that any conditions imposed upon $$u_{n}$$ may be violated for a finite number of terms: all that is necessary is that such a condition (e.g. that all the terms are positive) should be satisfied from some definite term onwards. Similarly in the case of an infinite integral the conditions have been supposed to be satisfied for all values of $$x$$ greater than some definite value, or for all values of $$x$$ within some definite interval $${[a, a + \delta]}$$ which includes the value $$a$$ near which the subject of integration tends to infinity. Thus our tests apply to such a series as $\sum \frac{n^{2} – 10}{n^{4}},$ since $$n^{2} – 10 > 0$$ when $$n \geq 4$$, and to such integrals as $\int_{1}^{\infty} \frac{3x – 7}{(x + 1)^{3}}\, dx,\quad \int_{0}^{1} \frac{1 – 2x}{\sqrt{x}}\, dx,$ since $$3x – 7 > 0$$ when $$x > \frac{7}{3}$$, and $$1 – 2x > 0$$ when $$0 < x < \frac{1}{2}$$.

But when the changes of sign of $$u_{n}$$ persist throughout the series, i.e. when the number of both positive and negative terms is infinite, as in the series $$1 – \frac{1}{2} + \frac{1}{3} – \frac{1}{4} + \dots$$; or when $$\phi(x)$$ continually changes sign as $$x \to \infty$$, as in the integral $\int_{1}^{\infty} \frac{\sin x}{x^{s}}\, dx,$ or as $$x \to a$$, where $$a$$ is a point of discontinuity of $$\phi(x)$$, as in the integral $\int_{a}^{A} \sin\left(\frac{1}{x – a}\right) \frac{dx}{x – a};$ then the problem of discussing convergence or divergence becomes more difficult. For now we have to consider the possibility of oscillation as well as of convergence or divergence.

We shall not, in this volume, have to consider the more general problem for integrals. But we shall, in the ensuing chapters, have to consider certain simple examples of series containing an infinite number of both positive and negative terms.