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from which we see that although the generating function

may be used to obtain as many terms of the series as we please, it can be regarded as the true equivalent to the infinite series

only if the remainder

vanishes when n is indefinitely increased ; in other words only when the series is convergent.

522. The General Term. When the generating function can be expressed as a group of partial fractions the general term of a recurring series may be easily found.

Ex. Find the generating function, and the general term, of the recurring series

$1-7x-x^{3}-43x^{3}-\ldots$

Let the scale of relation be $1-px-qx^{2}$; then

$-1+7p-q=0,-43+p+7q=0$;

whence $p=1,q=6$; and the scale of relation is

$1-x-6x^{2}$.

Let S denote the sum of the series; then

which is the generating function.

If we separate ${\frac {1-8x}{1-x-6x^{2}}}$ into partial fractions, we obtain