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

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137.Fundamental test for convergence. Summing up first and then terms of a series, we have

(A) ,
(B) ,

Subtracting (A) from (B),

(C) .

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

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

(D) .

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

;

or, what amounts to the same thing,

(E) .

Hence, if the general (or nth) term of a series does not approach zero as 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:

(F) ,
(G) ,