Sequences Flashcards
1
Q
What is the definition of convergence?
A
∀ε > 0 ∃N∈N(aturals) s.t. ∀n ∈N(aturals), n >= N ⇒ |xn −x| <= ε.
2
Q
When does (xn)n tend to infinity?
A
∀K∈R(eals) ∃N ∈N(aturals) s.t. ∀n ∈N, n >= N ⇒ xn >= K.
3
Q
When does (xn)n tend to minus infinity?
A
∀K∈R(eals) ∃N ∈N(aturals) s.t. ∀n ∈N, n >= N ⇒ xn <= K.
4
Q
How do you lay out convergence questions?
A
Let ε > 0. Choose N …
Then for n >= N:
|xn - x| = …
5
Q
What type of sequences are bounded?
A
Convergent sequences
6
Q
What is the Monotone Convergence Theorem?
A
if a monotone sequence is convergent then it is bounded and vice versa
7
Q
What is the Bolzano-Weierstrass Theorem?
A
every bounded sequence xn has a subsequence that converges to x
