Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/240



Summing up first $$n$$ and then $$n + p$$ terms of a series, we have Subtracting (A) from (B),

From Theorem III we know that the necessary and sufficient condition that the series shall be convergent is that

for every value of $$p$$. But this is the same as the left-hand member of (C); therefore from the right-hand member the condition may also be written

Since (D) is true for every value of $$p$$, then, letting $$p = 1$$, a necessary condition for convergence is that

or, what amounts to the same thing,

Hence, if the general (or nth) term of a series does not approach zero as $$n$$ approaches infinity, we know at once that the series is non- convergent and we need proceed no further. However, (D) is not a sufficient condition; that is, even if the nth term does approach zero, we cannot state positively that the series is convergent ; for, consider the harmonic series

Here

that is, condition (E) is fulfilled. Yet we may show that the harmonic series is not convergent by the following comparison: