Chapter 2 (part b)

Question 1

Theorem- Subsequences of a convergent sequence converge to…

Answer 1

to the same limit as the original sequence

Question 2

Bolzano-Weierstrass Theorem

Answer 2

Every bounded sequence contains a convergent subsequence

Question 3

Cauchy Sequence- A sequence (a_n) is called a Cauchy sequence if…

Answer 3

for every ε > 0, there exists an N in N such that whenever n,m > N, it follows that |a_n - a_m|

Question 4

Every bounded sequence contains a convergent subsequence

Answer 4

Bolzano-Weierstrass Theorem

Question 5

for every ε > 0, there exists an N in N such that whenever n,m > N, it follows that |a_n - a_m|

Answer 5

Cauchy Sequence- A sequence (a_n) is called a Cauchy sequence if…

Question 6

Cauchy Convergence- A sequence (a_n) converges to a real number a if…

Answer 6

for every ε > 0, there exists an N in N such that whenever n > N it follows that |a_n - a|.

Question 7

for every ε > 0, there exists an N in N such that whenever n > N it follows that |a_n - a|.

Answer 7

Cauchy Convergence- A sequence (a_n) converges to a real number a if…

Question 8

Cauchy Criterion

Answer 8

A sequence converges if and only if it is a Cauchy sequence.

Question 9

A sequence converges if and only if it is a Cauchy sequence.

Answer 9

Cauchy Criterion

Question 10

Algebraic Limit Theorem for Series-

If Σa_k = A and Σb_k = B, then…

Answer 10

i. ) Σca_k = cA for all c in R
ii. ) Σ(a_k + b_k) = A + B

Question 11

i. ) Σca_k = cA for all c in R
ii. ) Σ(a_k + b_k) = A + B

Answer 11

Algebraic Limit Theorem for Series-

If Σa_k = A and Σb_k = B, then…

Question 12

Cauchy Criterion for Series

The series Σa_k converges…

Answer 12

iff, given ε > 0, there exists an N in N such that whenever n > m > N it follows that

|a_m+1 + a_m+2 + …. + a_n|

Question 13

iff, given ε > 0, there exists an N in N such that whenever n > m > N it follows that

|a_m+1 + a_m+2 + …. + a_n|

Answer 13

Cauchy Criterion for Series

The series Σa_k converges…

Question 14

Theorem- If the series Σa_k converges, the limit of (a_n)…

Answer 14

must be equal to 0.

Question 15

Comparison Test- Assume (a_k) and (b_k) are sequences satisfying 0 < a_k < b_k for all k in N, then…

Answer 15

i. ) If Σb_k converges, then Σa_k converges.
ii. ) If Σa_k diverges, then Σb_k diverges.

Question 16

Geometric Series- A series is called geometric if it…

Answer 16

is of the the form

Σar^k = a + ar + ar² + ….

and converges if and only if

|r| < 1

Question 17

i. ) If Σb_k converges, then Σa_k converges.
ii. ) If Σa_k diverges, then Σb_k diverges.

Answer 17

Comparison Test- Assume (a_k) and (b_k) are sequences satisfying 0 < a_k < b_k for all k in N, then…

Question 18

is of the the form

Σar^k = a + ar + ar² + ….

and converges if and only if

|r| < 1

Answer 18

Geometric Series- A series is called geometric if it…

Question 19

Absolute Convergence Test- If the series Σ|a_n| converges, then…

Answer 19

the Σa_n converges as well.

Question 20

Alternating Series Test- Let (a_n) be a sequence satisfying

i. ) a₁ > …..
ii. ) limit (a_n) goes to 0
then. ..

Answer 20

the alternating series

Σ(-1)ⁿ⁺¹a_n

converges.

Question 21

Theorem- If a series converges absolutely, then…

Answer 21

any rearrangement of the series converges to the same limit.

Question 22

any rearrangement of the series converges to the same limit.

Answer 22

Theorem- If a series converges absolutely, then…

Question 23

Ratio Test- Given a series Σa_n with a_n not equal to 0, then if…

Answer 23

lim|a_n+1 / a_n| = r < 1

then the series converges absolutely.