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is too much by; four terms give , which is too small by, and so on.
181. The fraction may also be resolved into
an infinite series another way; namely, by dividing 1 by a+ 1,as follows:
It is however unnecessary to carry the actual division any farther, as we are enabled already to continue the series to any length, from the law which may be observed in those terms we have obtained; the signs are alternately plus and minus, and each term is equal to the preceding one multi
plied by "
It is thus that by changing the order of the terms of the denominator, we obtain the quotient under different forms, and that we pass from a diverging series, for certain values of a, to a converging series for the same values.
It may also be here observed, that in the division of the two plynomials, if we deviate from the established rule (Art. 93), we arrive at quotients which do not terminate :.
Thus, for example, a2b2, divided by a+b, according to the rule above quoted, gives for the quotient a-b; but if we divide a2-b2 by b+a, we shall arrive at a quotient which does not terminate:
Here, we can clearly see that the quotient will not terminate, however far we may continue the operation, because we have always a remainder.
In this case, by taking b+a for a divisor, we must, in order to find the quotient a-b, divide the whole dividend by all the divisor, that is to say, a2 —b2 or (a+b)× (a−b) by a+b.
182. When there are more than two terms in the divisor, we may also continue the division to infinity in the same manner.
Ex. 4. It is required to convert
an infinite series.
+a+a, &c. to infinity: where, if we make a=1,
=1=1+1—1—1+1+1, &c., which series contains twice the series found, (Art. 178), 1-1+1-1+1, &c. Now, as we have found this to be equal to, it is not extraordinary that we should find, or 1, for the value of that which we have just determined.
By making a=, we shall have÷=1=1+1-! 16 +6 +128512, &c.
If a=1, we shall have
And if we take the four leading terms of this se• ries, we have 4, which is only less than .
Let us suppose again a, and we shall have
} = ÷ 2 = 1 + 3 — 24 — i++, &c. this series is therefore equal to the preceding one, and by subtracting one from the other, we obtain -+,&c., which is necessarily =0.
183. The method which has been here explained, serves to resolve, generally, all fractions into infinite series; which is often found, as has been observed by EULER in his Algebra, to be of the greatest utility; it is also remarkable, that an infinite series, though it never ceases, may have a determinate value. It should likewise be observed, that from this branch of Mathematics, inventions of the utmost importance have been derived, on which account, the subject deserves to be studied with the greatest attention,