Calculus: Analytic Geometry and Calculus, with Vectors

12.1 Definitions and basic theorems 593

4 Considering separately the cases in which x = 0, 0 < x < 1, and -1 <
x < 0, use one orthe other or both of the formulas

1 -xn+i

= 1+x+x2+ +xn, x=1+x+x2+x3+

1-x 1-

to determine the nature of the sequence of partial sums of the geometric series.

5 Prove that if lakl =< 1 for each k, then the series

ao + aix + a2x2 + a3x3 + ...

converges when Ixl < 1. Hint: Use a dominating series.

6 The preceding problem is important and must be thoroughly understood.

Tell what is meant by the statement that the series

(1) ao+alx+azx2+a3x3+

is dominated by the series

(2) 1+1x1+1x12+1x13+ ...

Tell why the series (2) converges when lxi < 1. Give a full statement of the

comparison test for convergence of series. Remark: We should hear very often

that (1) is called a power series, that the numbers ao, a,, a2,... are called

constants, and that the number x is called a variable. We should not, however,

allow the terminology to interfere with our understanding of Problem S.

7 Prove that if 11 and p are positive constants for which

then the series

laki < 4pk

ao + aix + a2x2 + a3x3 + ...

converges when lxl < 1/p. Solution: In the stated circumstances lakxkI =

11lpxlk and the series is dominated by the convergent series in

1

1 - Ipxl =

11 + A'1 px1 + 111 pxl2 + 111 pxl3 +. ...

8 Prove that if the series

CO+ Clx + C2x2 + C3x3 +

converges when x = xo, then it also converges when lxi < lxol. Solution: This

problem is much like the preceding one. The hypothesis implies that lim ckxo

= 0 and hence that there is a constant M for which

Ickxol < M (k= 0, 1, 2,

If lxl < lxol, then

1

l ckxi < MI x/xol k (k = 0, 1, 2, ...)

and the series is dominated by a convergent geometric series.

9 Show that

I =^112 +^213 +

34

+

4.5

• 516 + ...