Chapter 2 (part a)

Question 1

A sequence (a_n) converges to a real

number a if, for every ε > 0, …

Answer 1

there exists an N in N such that whenever

n > N it follows that

|a_n - a| < ε

Question 2

there exists an N in N such that whenever

n > N it follows that

|a_n - a| < ε

Answer 2

A sequence (a_n) converges to a real

number a if, for every ε > 0, …

Question 3

Epsilon Neighborhood

Given a real number a in R and

and ε > 0 the set…

Answer 3

V_ε = {x in R: |x - a| < ε}

is the ε neighborhood of a.

Question 4

A sequence (a_n) converges to a

if, given any ε neighborhood V_ε of a,

there exists…

Answer 4

a point in the sequence after which

all terms are in V_ε(a).

(Every ε neighborhood contains all

but a finite number of the terms

of (a_n).)

Question 5

a point in the sequence after which

all terms are in V_ε(a).

(Every ε neighborhood contains all

but a finite number of the terms

of (a_n).)

Answer 5

A sequence (a_n) converges to a

if, given any ε neighborhood V_ε of a,

there exists…

Question 6

The limit of a sequence, when it exists…

Answer 6

must be unique.

Question 7

A sequence (x_n) is bounded if there exists

Answer 7

a number M > 0 such that

|x_n| M

for all n in N.

(In other words, there is an interval

[-M,M] that contains every term in a sequence]

Question 8

must be unique.

Answer 8

The limit of a sequence, when it exists…

Question 9

a number M > 0 such that

|x_n| < M

for all n in N.

(In other words, there is an interval

[-M,M] that contains every term in a sequence]

Answer 9

A sequence (x_n) is bounded if there exists…

Question 10

Algebraic Limit Theorem

Let lim (a_n) = a and lim (b_n) = b,

then…

Answer 10

i. ) lim (ca_n) = c lim (a_n)
ii. ) lim (a_n + b_n) = a + b
iii. ) lim (a_nb_n) = ab
iv. ) lim (a_n/b_n) = a/b

Question 11

i. ) lim (ca_n) = c lim (a_n)
ii. ) lim (a_n + b_n) = a + b
iii. ) lim (a_nb_n) = ab
iv. ) lim (a_n/b_n) = a/b

Answer 11

Algebraic Limit Theorem

Let lim (a_n) = a and lim (b_n) = b,

then…

Question 12

Order Limit Theorem

Assume lim (a_n) = a and lim (b_n) = b,

then…

Answer 12

i. ) If a_n > 0 for all n in N, then a_n > 0.
ii. ) If a_n < b_n for all n in N, then a < b.
iii. ) If there exists a c in R for which

c < b_n for all n in N, then c < b_._

iv.) Similarly, if a_n < c for all n in N,

then a < c.

Question 13

i. ) If a_n > 0 for all n in N, then a_n > 0.
ii. ) If a_n < b_n for all n in N, then a < b.
iii. ) If there exists a c in R for which

c < b_n for all n in N, then c < b.

iv.) Similarly, if a_n < c for all n in N,

then a < c.

Answer 13

Order Limit Theorem

Assume lim (a_n) = a and lim (b_n) = b,

then…

Question 14

Squeeze Theorem

If x_n < y_n < z_n

for all n in N, and if

lim(x_n) = lim(y_n) = L, then…

Answer 14

lim(y_n) = L

Question 15

Cesaro Means Theorem

If (x_n) converges to some limit L, then the sequence of the averages….

Answer 15

y_n = (x₁ + x₂ + … + x_n)/n

converges to the same limit L.

Question 16

y_n = (x₁ + x₂ + … + x_n)/n

converges to the same limit L.

Answer 16

Cesaro Means Theorem

If (x_n) converges to some limit L, then the sequence of the averages….

Question 17

Arithmetic and Geometric Mean

For any x,y > 0, the geometric mean is less than or equal to the arithmetic mean, or…

Answer 17

(xy)^(1/2) < (x + y)/2

Question 18

A sequence (a_n) is increasing if…

Answer 18

a_n< a_n+1

for all n in N

Question 19

a_n< a_n+1

for all n in N

Answer 19

A sequence (a_n) is increasing if…

Question 20

A sequence (a_n) is decreasing if…

Answer 20

b_n > b_n+1

for all n in N

Question 21

b_n > b_n+1

for all n in N

Answer 21

A sequence (a_n) is decreasing if…

Question 22

A sequence is montone if…

Answer 22

it is either increasing or decreasing.

Question 23

Monotone Convergence Theorem

A sequence converges if…

Answer 23

it is monotone and bounded.

Question 24

Let (b_n) be a sequence. An infinite series is a formal expression in the form…

Answer 24

Σ_{n=1 to inf} (b_n)

= b₁ + b₂ + ……

Question 25

Σ_{n=1 to inf} (b_n)

= b₁ + b₂ + ……

Answer 25

Let (b_n) be a sequence. An infinite series is a formal expression in the form…

Question 26

The sequence of partial sums (S_m) of an infinite series is…

Answer 26

the sequence (S_m) where

S_m = a₁ + a₂ + … + a_m

Question 27

the sequence (S_m) where

S_m = a₁ + a₂ + … + a_m

Answer 27

The sequence of partial sums (S_m) of an infinite series is…

Question 28

Theorem- An infinite series converges to L if…

Answer 28

the sequence of partial sums (S_m) converges to L.

Then, Σ_{n=1 to inf} (a_n) = L.

Question 29

the sequence of partial sums (S_m) converges to L.

Then, Σ_{n=1 to inf} (a_n) = L.

Answer 29

Theorem- An infinite series converges to L if…