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
Let the scale of relation be ; then
;
whence ; and the scale of relation is
.
Let S denote the sum of the series; then
which is the generating function.
If we separate into partial fractions, we obtain