Page:Elementary algebra (1896).djvu/434

416 Rh Hence ,

for the coefficient of every other power of @ is zero in consequence of the relation

Thus the sum of a recurring series is a fraction whose denominator is the scale of relation.

520. If the second fraction in the result of the last article decreases indefinitely as n increases indefinitely, the formula for the sum of an infinite number of terms of a recurring series of the second order reduces to

If we develop this fraction in ascending powers of x as explained in Art. 487, we shall obtain as many terms of the original series as we please; for this reason the expression

is called the generating. function of the series. The summation of the series is the finding of this generating function.

If the series is of the third order,

521. From the result of Art. 519, we obtain