Analysis I (Chapters 1-3) Flashcards
ch1: sequences ch2: series ch3: functions of a real variable
State the Archimedean property.
let x∈ℝ. Then there exsists n∈ℕ such that n>x.
What properties does a non-decreasing sequence have?
∀n∈ℕ: a(n)≤a(n+1)
What properties does a non-increasing sequence have?
∀n∈ℕ: a(n)≥a(n+1)
define a convergent sequence
a sequence a(n) converges to l as n→∞if
∀ε>0 ∃N∈ℕ s.t. ∀n∈ℕ(≥N): | a(n)-l |
TRUE OR FALSE:
A sequence converges if and only if it is Cauchy
TRUE
Define a bounded sequence
a(n) is bounded if ∃a,b∈ℝ, ∀n∈ℕ: a≤a(n)≤b
TRUE OR FALSE:
All bounded sequences are convergent
FALSE
all convergent sequences are bounded
what is the monotone convergence theorem?
Let a(n) be bounded from above and non-decreasing, and b(n) bounded from below and non-increasing. Then a(n)→sup(a(n)) and b(n)→inf(b(n)).
what is the comparison theorem?
Let a(n) and b(n) be sequences satisfying b(n)→0 and ∀n∈ℕ: |a(n)|≤b(n) then a(n)→0
what is the sandwich theorem?
Let a(n), b(n) be sequences such that a(n)→x, b(n)→x (where x∈ℝ) then ∀n∈ℕ: a(n)≤c(n)≤b(n). Then c(n)→x.
define a limit point/accumulation point
let a(n) be a sequence and c∈ℝ. Then c is an accumulation point iff there exists a subsequence a(nk) of a(n) such that a(nk)→c.
What is the Bolano-Weierstrass theorem?
Let a(n) be a bounded sequence and a,b∈ℝ such that ∀n∈ℕ: a(n)∈[a,b]. Then a(n) contains a convergent subsequence. In particular, a(n) has an accumulation point in [a,b].
TRUE OR FALSE:
Every unbounded sequence must contain a subsequence that converges to ∞ or -∞
TRUE
Let ∑a(n) be convergent. Then what can be said about a(n)?
a(n)→0
What does it mean to say a series is absolutely convergent?
A series ∑a(n) is absolutely convergent iff the series ∑|a(n)| converges.
what is the comparison theorem?
Let ∑a(n) and ∑b(n) be series and |a(n)|≤b(n) and ∑b(n) convergent. then ∑a(n) is absolutely convergent.
what is the integral comparison theorem?
let f be non-increasing. Then ∑f(n) (from n=1 to ∞) converges iff ∫f(x) dx (from 1 to ∞) converges.
What is the root test?
Let a(n) be a sequence and a:= limsup(n→∞) (|a(n)|^1/n). Then ∑a(n) is absolutely convergent if a<1 and ∑a(n) is divergent if a>1.
what is the ratio test?
Let a(n) be a sequence with ∀n∈ℕ: a(n)≠0, and limsup(|a(n+1)/a(n)|<1. Then, ∑a(n) converges absolutely. If lim|a(n+1)/a(n)|>1, then ∑a(n) diverges.
what is conditional convergence?
When a series is convergent, but not absolutely so.
Define Power series and Radius of convergence.
Let c(n) be a sequence and x∈ℝ. Then, ∑c(n)x^n is called a power series and its radius of convergence is R:= (limsup |c(n)|^1/n)^-1.
Let ∑c(n)x^n be a power series and R its radius of convergence. Then what can be said about ∑c(n)x^n for
(a) |x|R
(a) absolutely convergent
(b) divergent
Let ∑C(n)x^n be a power series with radius of convergence R. Given that c(n)≠0 and |c(n)/c(n+1)|→C. Then what can be said of R?
R=C
When is a FUNCTION bounded?
when its range is bounded
define the limit of f at the point x
u is called the limit of f at the point x (lim(y→x) f(y)=u)iff ∀ε>0 ∃δ>0 ∀y∈I{x}: |y-x|
when is a function continuous at a point?
When the left limits and right limits at that point exist and coincide with f(x)
If the sequence a(n) does not converge, what can be said of the corresponding series?
It also does not converge
A function f is continuous on the open interval (-1,1). is the function made up of the union of the open intervals (-1,0) and (0,1) continuous?
No. f(a)>0 and f(b)<0 for some a,b∈(-1,1). By the intermediate value theorem, there is a point c lying between a and b such that f(c)=0
