Calculus: Analytic Geometry and Calculus, with Vectors

622 Series

For example, the series in (12.412), (12.413), and (12.414) are the only

power series in x that converge to ex, cos x, and sin x. To prove this

theorem, we can start with (12.482) and show that bk = f(k)(a)/k! just

as we started with (12.43) and showed that Ck = f(k)(a)/k!.

Problems 12.49

1 Learn the formulas

x2 x3 x4

(a) ex+x+22+33+,x+

3 5

(b) sinx=x3i+ 5 X7

x2 x4 x5

(c) cosx=1-2i+4-6i+ '

X +X+x2+x3+

Write the four formulas obtained by differentiating formulas (a) to (d).

2 Explain the steps by which the series

1-t=1+t+t2+t3+

and modifications of it can be used to obtain the formulas

(a) 1+x=1 -x+x2-x3+ ... (Ixl <1)

(b) log (1

+X)

= x - 2 +

33

• 4 + ... (Ixl < 1)

(c) 1 + x =

1 - x2 + x4 - x5 + ... (Ixl < 1)

(d) tan-' x = x -x3 x5

3

+ xT

S

• 7

+.. (Ixl < 1)

(e) (1 x)2 = 1 + 2x + 3x2 + 4x3 + (ixl < 1)

(f) limlog (I + x)= 1' limx - log (1 - x)=^1

X- O X x-.0 x2 2

f

s

(8) log (I +t)

o

t

dt=x-x2 +x3 -x4

223242 + Qxl <1)

3 We can object to the general principle that problems should be solved in

inefficient ways, but nevertheless we can sometimes profitably sacrifice a few

square feet of paper to promote understanding of a subject. Assuming that the

series in

sin 2x= CO + C'x + c2x2 + C3x3 + C4X4 + ..

converges to sin 2x, find c0, c', c2, in the following way. Put x = 0 to

find ca. Differentiate once and put x = 0 to find c1. Differentiate once more