UNDETERMINED COEFFICIENTS
393
Equating the coefficients of like powers of
, we have
thus \sqrt{1+x} = 1+\frac{x}{2}} +\frac{x^2}{8} ++\frac{x^3}{16} - \frac{5x^4}{128} + \cdots.
Note, The expansion can be readily effected by the use of the Binomial Theorem [Art. 421].
Expand the following expressions to four terms :
1.
.
2.
.
3.
.
4.
.
5.
.
6.
.
REVERSION OF SERIES.
490. To revert a series
is to express
in a series of ascending powers of
.
Revert the series
|
(1)
|
Assume x = Ay+ By^2 + Cy^3 + Dy^4 + .
Substituting in this equation the value of y as given in
(1), we have
Equating the coefficients of like powers of x,
Hence
.