Calculus: Analytic Geometry and Calculus, with Vectors

12.3 Alternating series and Fourier series 615

2 Find, correct to four decimal places, the numbers to which the following
series are convergent.

1 2 3 4 5

(a) 10 102 + 103 104 + 105

12 22 32 42 52

(b) 10 102 + 103 104 + 105 -

...

3 Show that the series

2 3 4

for which the nth term is xn/(2n + 1), converges when -1 5 x < 1 and di\erges

when x < -1 and N%hen x > 1. Hznt: Some but not all of the information is

revealed by the ratio test.

4 With the aid of basic information about alternating series show that the

series in

(1)

1

5 =

1- 1

-I

1

1

(^1222) 32T2-i 52-
converges to a number S for which 0 < S < 1. Then show that, correct to 5D
(5 decimal places),
S > 0.75000 S < 0.86111 S > 0 79861 S < 0.83861
Remark: One who wishes to invest a moment to pick up some ideas may start
with the esoteric but important formula
2
6 12 22 32 42 52 62
and obtain the formula
(3)
2
24-0+22+0+42+0+62+
Subtracting twice (3) from (2) then gives (1) with S = ar2/12. Subtracting
(3) from (2) gives the formula
2

(4) s = 12-1-32+s2+32+92+

which sparkles almost as brightly as (2).

5 Supposing that 0 < x < 1, use the formula

log(1+x)=x-

x2

2 +

x33

x44 + ...

to show that

and

z^3

x-log(1+X)=x202 +4 5+

0 < x - log (1 + x) < 'x2.