...whatever may be the value of x, then by bringing the terras all to one side, we infer that that is, the coefficients of like powers of x in the two series are equal. The theorem here given is sometimes quoted as the Principle of Indeterminate Coefficients; we assumed its...

......... are always equal whatever may be the value of x, then ^-A0+(a1-Al)x+(a1-As)xt+ ...... that is, the coefficients of like powers of x in the two series are equal. The theorem here given is sometimes quoted as the Principle of Indeterminate Coefficients; we assumed its...

...convergent these two series approach the SAME limit as the number of terms increases, without limit, then the coefficients of like powers of x in the two series are equal each to each. Since we are dealing with the limit of convergent series as the number of terms increases...

...series A + Bx + CV4 ----- h- Kx" and A' + B'x + C'x--\ ----- \-K'x", are equal for every value of x, the coefficients of like powers of x in the two series are equal. ALGEBRA. APPLICATION TO THE EXPANSION OF FRACTIONS INTO SERIES. 378. 1. Esp&ud 2-Зaг-x3 1 — 2ж+3ж2...

...has any value which makes both members convergent, and by the Theorem of Undetermined Coefficients, the coefficients of like powers of x in the two series are equal ; that is, A= 2. B — 2A= 0; whence, B= 2 A = 4. -3; whence, C = 2B-3A-3=-l. -1; whence, D=2C-3B-l=-15....

...+ B'x + C'x2 + D'x3 + • ••, where x has any finite value which makes both series convergent, the coefficients of like powers of x, in the two series, are the same. The above method is called The Theorem of Undetermined Coefficients. Notes: 1. The equation...

...or, f(jc) = <v» - (6, - KJ-r"-1 -f (b, - hb, )j— ' + .. . +R- Aft..,. (11) These expressions being identically equal, the coefficients of like powers of x in the two members must be equal ; therefore, na =/<„,«,= b, /(«„, a, == 62 — /i6, , . . . an = R-hbn_,....

...conditions since it is equal to series (5). Since the two expansions of a'+". are identically equal, then the coefficients of like powers of x in the two series are equal, thus (6) Al + 2Af4 + 3A3y* + iAty>+ .... [1669] = ^«• = ^(1 + ^+^«+^+ ....). These series are convergent...

...A<l+Alx+A^+ .... and B^-lB^-^-BfC1-^- .... are equal for all values of x for which they are convergent, then the coefficients of like powers of x in the two series are separately equal. For the values of ./• for which the two series are always convergent they are finite...

...Theorems respectively, \ve shall obtain two convergent infinite series in ascending powers of x. Therefore the coefficients of like powers of x in the two series are equal. Art. (314). It is, therefore, only necessary to obtain the coefficient of xn in the right hand member....