Calculus: Analytic Geometry and Calculus, with Vectors

12.4 Power series 623

and put x = 0 to find c2. Continue the process until c5 has been obtained.
Finally, see whether the result agrees (as it should) with the result of replacingx
by 2x in the basic formula
x3 x5 x7

(^4) Repeat the operation of Problem 3 to find the expansion of ebr in powers
of x, it being assumed that b > 0 and that there is a power series in x thatcon-
verges to ebz. Tell how your answer can be checked.
5 Assuming that there exist constants co, c1, c2, for which
f(x)
do enough differentiating and substituting to find the first few c's when
(a) f (x) _ (1 - x)-1, a = 0
(b) f(x) = x-1, a = 1
(c) f(x) = log (1 + x), a = 0
(d) f (x) = log x, a = 1
6 Without bothering to write derivatives of the right member of the formula

f(x) =co+cl(x-a)+c2(x-a)2+c3(x-a)3-+.. .,

suppose that the series converges to f(x) and find the first few of the c's with the

aid of the formula

f(x) = f(a) +tLa) 1 i(x - a) + f 2 a) (x - a) 2 + 3 ia) (x- a)3 + ...

when

(a) f(x) = sin x, a = 0

(b) f (x) _ (1 - x)-1, a = 0

(c) f(x) _ (1 + x)3, a = 0

(d) f (x) = x3 - 2x2 + x - 1, a = 1

7 It is possible to apply the methods of the preceding problems to calculate

a few coefficients in cases where the complexities of formulas for f(")(x) increase

very rapidly as n increases. In some such cases, it is worthwhile to know the

numerical values of the first one or two or three nonzero coefficients. Verify

the first two nonzero coefficients in each of the formulas

(a) tan x =x+3x3+rsxs+Mx7+ yH-SX9 ...

a

(XI <2!

(b) secx =1 +3x2+Ax4+- 2ax6 +sil-x1 + ... <2

(c) (1+x+x2)'=1+3x+XIx2+ ...

(^8) It can be proved that if the series in the first two formulas
(1) f(x) = ao + aIx + a2x2 + asx3 + a4x4 +
(2) g(x) = bo + bix + b2x2 + bax3 + b4x4 .+ ...
(3) f(x)g(x) = co + c1x + c2X2 + cax3 + C4X4 + ...