Page:Elementary algebra (1896).djvu/436

By actual division, or by the Binomial Theorem.

${\frac {2}{1+2x}}=2[1-2x+(2x)^{2}-\ldots +(-1)^{r}-(2x)^{r}+\ldots ]$ ;

$-{\frac {1}{1-3x}}=-[1+3x+(3x)^{2}+\ldots +(3x)^{r}+\ldots ]$

Whence the (r+1)th, or general term, is

$[2(2^{r})(-1)^{r}-3^{r}]x^{r}={(-1)^{r}-2^{r+1}-3^{r}}x^{r}$.

EXAMPLES XLIV. a.

Find the generating functions of the following series.

1. 1 + 6 X + 24 x2 + 84 x3 + •••• 2. 2 + 2 X - 2 x2 + 6 x3 - 14 x"* + - ••• 3. 3 - 16 x + 42x2-94x3+ .-.. 4. 2 - 5 X + 4 x2 + 7 x3 - 26 x* + .-. 5. 4 + 5x + 7x2 + 11x3+ .... 6. 1 + x + 2x2 + 2x3 + 3x'i + 3x5 + 4x6 + 4x7+ ••.. 7. 1 + 3x + 7x2+ 13x3 + 21x4 + 31x5+ .... 8. 1 -3x + 5x2- 7x3 + 9x*- 11x5 + ....

Find the generating function and the general term in each of the following series :

9. 1 + 5x + 9x2+ 13x3+ .... 11. 2 + 3x + 5x2 + 9x3+ .... 10. 2-x + 5x2-7x3+ .... 12. 7 -6x + 9x2 + 27x4 + .... 13. 3 + 6x + 14 x2 + 36 x3 + 98 x^ + 276 x^ + ....

THE METHOD OF DIFFERENCES.

523. Let $u_{n}$ denote some rational integral function of n, and let $u_{1},u_{2},u_{3},u_{4},\ldots$ denote the values of $u_{n}$ when for n the values 1, 2, 3, 4, are written successively.

From the series $u_{1},u_{2},u_{3},u_{4},u_{5}\ldots$ obtain a second series by subtracting each term from the term which immediately follows it.

The series found is called the series of the first order of differences, and may be conveniently denoted by $Du_{1},Du_{2},Du_{3},Du_{4},\ldots$ .

By subtracting each term of this series from the term that immediately follows it, we have Du^ — Dui, Du^ — Du2,