Sequences

Question 1

What is the definition of convergence?

Answer 1

∀ε > 0 ∃N∈N(aturals) s.t. ∀n ∈N(aturals), n >= N ⇒ |xn −x| <= ε.

Question 2

When does (xn)n tend to infinity?

Answer 2

∀K∈R(eals) ∃N ∈N(aturals) s.t. ∀n ∈N, n >= N ⇒ xn >= K.

Question 3

When does (xn)n tend to minus infinity?

Answer 3

∀K∈R(eals) ∃N ∈N(aturals) s.t. ∀n ∈N, n >= N ⇒ xn <= K.

Question 4

How do you lay out convergence questions?

Answer 4

Let ε > 0. Choose N …
Then for n >= N:

|xn - x| = …

Question 5

What type of sequences are bounded?

Answer 5

Convergent sequences

Question 6

What is the Monotone Convergence Theorem?

Answer 6

if a monotone sequence is convergent then it is bounded and vice versa

Question 7

What is the Bolzano-Weierstrass Theorem?

Answer 7

every bounded sequence xn has a subsequence that converges to x