Sequences Fundamentals

Question 1

When does a sequence converge to a limit L = lim k tends to inf Ak

Answer 1

A sequence converges to a limit L = lim k tends to inf Ak when:
For every E > 0 E N e N s.t mod(Ak-L) < epsilon for all k >= N

Question 2

When does Ak diverge positively (to inf)

Answer 2

Ak diverges positively when:
For every M in R, E N e N s.t Ak > M for all k >=N

Question 3

When does Ak diverge negatively (to -inf)

Answer 3

Ak diverges negatively when:
Every M in R, E N e N s.t Ak < M for all k >= N

Question 4

What is Cauchy criterion for convergence

Answer 4

Cauchy criterion for convergence is:
Sequence Ak converges iff for each epsilon > 0, E N s.t
Mod(Ak - AL) < epsilon for all k, L, >= N

Question 5

What is lim k tends to inf A^k for a > 1, a = 1, -1 < a < 1. When does it not exist

Answer 5

Lim k tends to inf a^k for a > 1 is: inf
For a = 1, =1
For -1 < a < 1 , = 0
Doesn’t exist for a <= -1

Question 6

What is lim a^1/k k tends to inf (a > 0)

Answer 6

Lim a^1/k k tends to inf (a > 0) = 1

Question 7

What is lim k tends to inf k^b for b > 0 and b < 0

Answer 7

Lim k tends to inf k^b for b > 0 is inf
For b < 0 = 0

Question 8

What is lim k tends to inf k^b/k

Answer 8

Lim k tends to inf k^b/k = 1

Question 9

What is lim k tends to inf inf Ak

Answer 9

Lim k tends to inf inf Ak is lim(inf(Am : m >= k)) = sup(inf(Am : m >= k)) infimum of sequence after a long time

Question 10

What is lim k tends to inf sup Ak

Answer 10

Lim k tends to inf sup Ak is:
Lim k tends to inf(sup(Am : m >= k)) = inf(sup(Am : m >= k)) supremum after a long time