If x is numerically less than 1, the sum approaches to the and the series is therefore convergent.

finite limit

1

- X

If is numerically greater than 1, the sum of the first and by taking n sufficiently great, this can

n terms is

x" - 1

X 1

be made greater than any finite quantity; thus the series is divergent.

If x = 1, the sum of the first n terms is n, and therefore the series is divergent.

If x=1, the series becomes

1 − 1 + 1 − 1 + 1 − 1 +

The sum of an even number of terms is 0, while the sum of an odd number of terms is 1; and thus the sum oscillates between the values 0 and 1. This series belongs to a class which may be called oscillating or periodic convergent series.

279. There are many cases in which we have no method of finding the sum of the first n terms of a series. We proceed therefore to investigate rules by which we can test the convergency or divergency of a given series without effecting its summation.

280.

An infinite series in which the terms are alternately positive and negative is convergent if each term is numerically less than the preceding term.

The given series may be written in each of the following forms:

From (1) we see that the sum of any number of terms is a positive quantity; and from (2) that the sum of any number of terms is less than u,; hence the series is convergent.

each term is numerically less than the preceding term, and the series is therefore convergent. But the given series is the sum of

Now (1) is equal to log, 2, and (2) is equal to 0 or 1 according as the number of terms is even or odd. Hence the given series is convergent, and its sum continually approximates towards log. 2 if an even number of terms is taken, and towards 1 + log, 2 if an odd number is taken.

282. An infinite series in which all the terms are of the same sign is divergent if each term is greater than some finite quantity however small.

For if each term is greater than some finite quantity a, the sum of the first n terms is greater than na; and this, by taking n sufficiently great, can be made to exceed any finite quantity.

283. Before proceeding to investigate further tests of convergency and divergency, we shall lay down two important principles, which may almost be regarded as axioms.

I. If a series is convergent it will remain convergent, and if divergent it will remain divergent, when we add or remove any finite number of its terms; for the sum of these terms is a finite quantity.

II. If a series in which all the terms are positive is convergent, then the series is convergent when some or all of the terms are negative; for the sum is clearly greatest when all the terms have the same sign.

We shall suppose that all the terms are positive, unless the contrary is stated.

284. An infinite series is convergent if from and after some fixed term the ratio of each term to the preceding term is numerically less than some quantity which is itself numerically less than unity. Let the series beginning from the fixed term be denoted by

285. In the enunciation of the preceding article the student should notice the significance of the words "from and after a fixed term."

and by taking n sufficiently large we can make this ratio approximate to x as nearly as we please, and the ratio of each term to the preceding term will ultimately be x. Hence if x < 1 the series is convergent.

Here we have a case of a convergent series in which the terms may increase up to a certain point and then begin to decrease.

286. An infinite series in which all the terms are of the same sign is divergent if from and after some fixed term the ratio of each term to the preceding term is greater than unity, or equal to unity.

Let the fixed term be denoted by u,. If the ratio is equal to unity, each of the succeeding terms is equal to u1, and the sum of n terms is equal to nu1; hence the series is divergent.

If the ratio is greater than unity, each of the terms after the fixed term is greater than u1, and the sum of n terms is greater than nu,; hence the series is divergent.

287. In the practical application of these tests, to avoid having to ascertain the particular term after which each term is greater or less than the preceding term, it is convenient to find

the limit of when n is indefinitely increased; let this limit

น

be denoted by

7-1

If <1, the series is convergent. [Art. 284.]

If λ> 1, the series is divergent. [Art. 286.]

If λ=1, the series may be either convergent or divergent, and a further test will be required; for it may happen that <1 but continually approaching to 1 as its limit when n is indefinitely increased. In this case we cannot name any finite quantity which is itself less than 1 and yet greater than A. Hence the test of Art. 284 fails. If, however,

n

> 1 but con

tinually approaching to 1 as its limit, the series is divergent by

Example 1. Find whether the series whose nth term is

vergent or divergent.

= x;

a+(a + d) r+ (a +2d) r2 + ... + (a + n − 1 . d) pn−1 + ...,

thus if r<1 the series is convergent, and the sum is finite. [See Art. 60, Cor.]

288. If there are two infinite series in each of which all the terms are positive, and if the ratio of the corresponding terms in the two series is always finite, the two series are both convergent, or both divergent.

Let the two infinite series be denoted by

Hence if one series is finite in value, so is the other; if one series is infinite in value, so is the other; which proves the proposition.