Calculus 1, Quiz Instructor: Frank Liou
Exam Time: 20-30 min
Total Score: = min{Your score,20}.
1. Quiz 1
(1) (5 Points) Use mathematical induction to prove the following inequality 2n> n, n≥1.
Proof. Forn= 1,21 = 2>1.The statement is true for n= 1.
Suppose the statement is true for n=k,i.e. 2k> k.Then forn=k+ 1,we find 2k+1 = 2·2k >2k≥k+ 1.
This shows that the statement is also true forn=k+1.By mathematical induction, the inequality holds for all n≥1.
(2) Define a sequence of real numbers (an) recursively bys1 = 1/2 and
sn=sn−1+ 1
2n, n≥2.
(a) (5 Points) Find the limitsof (sn) using what you have learned in high school.
(b) (10 Points) Use definition to show that the result you find is correct, in other word, use definition to show that lim
n→∞sn = s. (You can either use the above inequality or prove it directly.)
Proof. It is not difficult to show that sn= 1− 1
2n, n≥1.
(Proof 1) Using (1), we know that for eachn≥1,
|sn−1|= 1 2n < 1
n.
For any >0,we chooseN = [1/] + 1.Then for n≥N,
|sn−1|< 1 n ≤ 1
N
< . Here we useN >1/. This proves that lim
n→∞sn= 1.
(Proof 2) For any >0,choose N = max{1,[−log2] + 1}.Then forn≥N, 2n≥2N >2−log2 = 1
.
1
2
Hence for n≥N,we have
|sn−1|= 1 2n <
using the above inequality.
(3) (Bonus: 10 Points) Letnbe a natural number. Show that
1 + 1
n+ 1> n+1 s
1 + 1
n n
.
(Hint: You may use the inequality of arithmetic and geometric means: suppose a1,· · · , an are nonnegative real numbers. Then
a1+· · ·+an
n ≥ √n
a1· · ·an. The equality holds if and only if a1 =· · ·=an.)
Proof. Leta1= 1 and a2=· · ·=an+1= 1 + 1/n.Then a1+· · ·+an+1
n+ 1 = 1 +n· 1 +n1
n+ 1 = n+ 2
n+ 1 = 1 + 1 n+ 1 and
a1· · ·an+1=
1 +1 n
n
.
The inequality is the direct consequence of the A.G. inequality.
