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380 Rh 473. Third Test. 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

and let

where r < 1.

Then

that is, < u_1 1-r, since r< 1.

Hence the given series is convergent.

474. In the enunciation of the preceding article the student should notice the significance of the words “from and after a fixed term.”

Consider the series

Here,

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 a. Hence if x<1, the series is convergent.

But the ratio u_n u_n-1 will not be less than 1, until nx n-1 <1; that is, until n > 1 1-x.

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