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391 Rh vanishes for all values of x within the assigned limits; therefore by the last article

that is,

hence the coefficients of like powers of the variable are equal, which proves the proposition.

EXPANSION OF FRACTIONS INTO SERIES.

487. Expand in a series of ascending powers of a.

Let

where A, B, C, D, ---, are constants whose values are to be determined; then

Equating the coefficients of like powers of 2, we have A=2, B+A=0, C+B-A=1, D+C-B=0, = B=-2; C=5; D =-7;

thus

488. Both numerator and denominator should be arranged with reference to the ascending powers of the same quantity; then dividing the first term of the numerator by the first term of the denominator determines the form of the expansion.