We will now show that it is convergent when that letter
equals zero.
Let the series be

a'{-hX'\-cx^'{-dx^+ -f A;a5'*--^+

The sum of the first n terms is

a'\-hX'\-cx^-\- dx^ -k- 4- /ca"-^,

which is equal to a, if x is made equal to 0.
Hence, however large n may be, the sum of the first n terms
is equal to a, if x is equal to 0. Consequently the series is
convergent if a; = 0.
411. Infinite series may be developed by the common pro-
cesses of Division, as in Art. 101, Exs. 19 and 20, and Extrac-
tion of Roots, as in Arts. 239 and 243; and by other methods
which it will now be our object to elucidate.
306 ALGEBRA.
UNDETERMINED COEFFICIENTS.
412. A method of expanding algebraic expressions into
series, simple in its principles, and general in its application,
is based on the following theorem, known as the



. THEOBEM OF TTHDETEBXINED COEFFICIENTS.
413. If the series A -\- Bx+ Cx'+ Dx^+ isalioays

equal to the series A' -^ B' x + C x^ -\- D' a^ + , for any

value ofx which makes both series convergent^ the coefficients
of like powers ofx in the two series will be equal.
For, since the equation
A + Bx-\- Cx^^-D^+ =A'-\-B'x+ (7ic«+ iXar* +



is satisfied for any value of x which makes both members con-
vergent ; and since by Art. 410, if x is equal to 0, both mem-
bers are convergent ; it follows that the equation is satisfied if
X = 0. Making x = 0, the equation becomes

A = A'.
