Test 1 9.1-9.4

Question 1

A ______ is a function whose domain is the set of positive integers.

Answer 1

sequence

Question 2

If f is the function and n is a positive integer, then

Answer 2

a_n = f(n)

Question 3

A sequence is ________ if either (a) if each term is greater than or equal to the preceding term, or (b) each term is less than or equal to the preceding term

Answer 3

monotonic

Question 4

A sequence {a_n} is ______ _______ if there is areal number N such that N ≤ a_n for all n.

Answer 4

bounded below

Question 5

The number M is called an _____ _____ for a sequence {a_n} if a_n ≤ M for all n.

Answer 5

upper bound

Question 6

The _____ of a sequence {a_n} is : if for each ℇ>0, there exists M>0 such that |a_n-L| < ℇ whenever n>M.

Answer 6

Limit

Question 7

In this case, the sequence _______ to L, and we write a_n → L

Answer 7

converges

Question 8

Assume that {a_n} converges to L and {b_n} converges to K.

If a sequence is bounded and monotonic then it (converges/diverges)

Answer 8

converges

Question 9

Assume that {a_n} converges to L and {b_n} converges to K.

{a_nb_n} converges to [(LK)/L+K]

Answer 9

L*K

Question 10

Assume that {a_n} converges to L and {b_n} converges to K.

{a_n/b_n} converges to (L/K / K/L), if b_n ≠ 0 and L≠0 and K≠0

Answer 10

L/K

Question 11

If {a_n} and {b_n} both converge to L, and if there exists and N such that for all n>N, a_n ≤ c_n ≤ b_n then {c_n} converges to L.

Answer 11

Squeeze Theorem

Question 12

The number N is called a _____ _____ for a sequence {a_n} if N≤a_n for all n.

Answer 12

lower bound

Question 13

A sequence {a_n} is _____ _____ if there is a real number M such that a_n ≤ M for all n.

Answer 13

bounded above

Question 14

A sequence is said to _______ if it has a limit.

Answer 14

converge

Question 15

A sequence is said to _______ if it does not have a limit.

Answer 15

diverge

Question 16

Assume that {a_n} converges to L and {b_n} converges to K.

If lim f(x) = M, as x→∞, and if c_n (= / ≠) f(n), then c_n → M.

Answer 16

=

Question 17

Assume that {a_n} converges to L and {b_n} converges to K.

{cb_n} converges to ( cK / c ), for any real number c.

Answer 17

c*K

Question 18

Assume that {a_n} converges to L and {b_n} converges to K.

{a_n +b_n} converges to (L+K / 2L).

Answer 18

L+K

Question 19

If | c_n | → (0 / M), then c_n converges.

Answer 19

0 (Absolute Value Theorem )

Question 20

For a sequence {a_n}, the ______ _____ has the form S_n= a1 +a2+a3+…+a_n

Answer 20

partial sum

Question 21

For any sequence {a_n}, the sequence {S_n} of its partial sums if called the ____ _____

Answer 21

series ∑a_n

Question 22

A series ________ if and only if its ________ _______ ______ converges; otherwise, the series _______.

Answer 22

Converges
sequence of parital sums
diverges

Question 23

The limit of a series is called its ____

Answer 23

sum

Question 24

A _____ ________ has the form (b1-b2)+(b2-b3)+(b3-b4)+….

Answer 24

telescoping series

Question 25

A _________ _________ has the form a∑r^n = a+ ar+ar^2 +a^3 + a^4….., with the index n going from 0 to ∞. The ______ is _ and the ____ ____ is _.

Answer 25

geometric series
ratio
r 
first time
a

Question 26

If the sequence of ( terms / partial sums) of {a_n} converges, then ∑a_n converges

Answer 26

partial sums

Question 27

If ∑a_n converges, then {a_n} converges to ( xero / a positive constant )

Answer 27

xero

Question 28

A geometric series with first a and ratio r will converge to a/(1-r), if |r| is ( greater / less ) than 1.

Answer 28

greater

Question 29

If {b_n} converges to a finite constant, B, then the telescoping series ∑(b_n -b_n+1) converges to ( B /b_1 -B )

Answer 29

B