Page 20 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 1] NUMBERS 11

1 1 1 1
Let S n ¼ þ þ þ þ
2 4 8 2 n 1
1 1 1 1 1
Then
2 S n ¼ 4 þ þ þ 2 n 1 þ 2 n
8
1 1 1 1
Subtracting, S n ¼ : Thus S n ¼ 1 < 1for all n:
2 2 2 n 2 n 1

EXPONENTS, ROOTS, AND LOGARITHMS
1.15. Evaluate each of the following:
4
3 3 8 3 4þ8 4þ8 14 2 1 1
ðaÞ 14 ¼ 14 ¼ 3 ¼ 3 ¼ 2 ¼
3 3 3 9
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
6 2 5 4 10 6 10 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð5 10 Þð4 10 Þ 2:5 10 9 25 10 10 ¼ 5 10 5 or 0:00005
8 10 8 10
ðbÞ 5 ¼ 5 ¼ ¼
3 3
2 3
2 x
27
27
log ¼ x: Then or x ¼ 3
ðcÞ 2=3 8 3 ¼ 8 ¼ 2 ¼ 3
ðdÞðlog bÞðlog aÞ¼ u: Then log b ¼ x; log a ¼ y assuming a; b > 0 and a; b 6¼ 1:
b
a
a
b
x
y
Then a ¼ b, b ¼ a and u ¼ xy.
x y
1
y
Since ða Þ ¼ a xy ¼ b ¼ a we have a xy ¼ a or xy ¼ 1the required value.
M
1.16. If M > 0, N > 0; and a > 0 but a 6¼ 1, prove that log a ¼ log M log N.
a
a
N
y
x
Let log M ¼ x,log N ¼ y. Then a ¼ M, a ¼ N and so
a
a
M a x M
¼ a x y or log ¼ x y ¼ log M log N
N ¼ a y a N a a
COUNTABILITY
1.17. Prove that the set of all rational numbers between 0 and 1 inclusive is countable.
1 2 3
Write all fractions with denominator 2, then 3; .. . considering equivalent fractions such as ; ; ; .. . no
2 4 6
more than once. Then the 1-1 correspondence with the natural numbers can be accomplished as follows:
Rational numbers 0 1 1 2 1 3 2 3 1 4 3 4 1 5 2 5 .. .
l lll lll ll
Natural numbers 1 2 3 4 5 6 7 8 9 .. .
Thus the set of all rational numbers between 0 and 1 inclusive is countable and has cardinal number F o
(see Page 5).
1.18. If A and B are two countable sets, prove that the set consisting of all elements from A or B (or
both) is also countable.
Since A is countable, there is a 1-1 correspondence between elements of A and the natural numbers so
that we can denote these elements by a 1 ; a 2 ; a 3 ; ... .
Similarly, we can denote the elements of B by b 1 ; b 2 ; b 3 ; ... .
Case 1: Suppose elements of A are all distinct from elements of B. Then the set consisting of elements from
A or B is countable, since we can establish the following 1-1 correspondence.