Analysis I (Chapters 1-3)

Question 1

State the Archimedean property.

Answer 1

let x∈ℝ. Then there exsists n∈ℕ such that n>x.

Question 2

What properties does a non-decreasing sequence have?

Answer 2

∀n∈ℕ: a(n)≤a(n+1)

Question 3

What properties does a non-increasing sequence have?

Answer 3

∀n∈ℕ: a(n)≥a(n+1)

Question 4

define a convergent sequence

Answer 4

a sequence a(n) converges to l as n→∞if

∀ε>0 ∃N∈ℕ s.t. ∀n∈ℕ(≥N): | a(n)-l |

Question 5

TRUE OR FALSE:

A sequence converges if and only if it is Cauchy

Answer 5

TRUE

Question 6

Define a bounded sequence

Answer 6

a(n) is bounded if ∃a,b∈ℝ, ∀n∈ℕ: a≤a(n)≤b

Question 7

TRUE OR FALSE:

All bounded sequences are convergent

Answer 7

FALSE

all convergent sequences are bounded

Question 8

what is the monotone convergence theorem?

Answer 8

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)).

Question 9

what is the comparison theorem?

Answer 9

Let a(n) and b(n) be sequences satisfying b(n)→0 and ∀n∈ℕ: |a(n)|≤b(n) then a(n)→0

Question 10

what is the sandwich theorem?

Answer 10

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.

Question 11

define a limit point/accumulation point

Answer 11

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.

Question 12

What is the Bolano-Weierstrass theorem?

Answer 12

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].

Question 13

TRUE OR FALSE:

Every unbounded sequence must contain a subsequence that converges to ∞ or -∞

Answer 13

TRUE

Question 14

Let ∑a(n) be convergent. Then what can be said about a(n)?

Answer 14

a(n)→0

Question 15

What does it mean to say a series is absolutely convergent?

Answer 15

A series ∑a(n) is absolutely convergent iff the series ∑|a(n)| converges.

Question 16

what is the comparison theorem?

Answer 16

Let ∑a(n) and ∑b(n) be series and |a(n)|≤b(n) and ∑b(n) convergent. then ∑a(n) is absolutely convergent.

Question 17

what is the integral comparison theorem?

Answer 17

let f be non-increasing. Then ∑f(n) (from n=1 to ∞) converges iff ∫f(x) dx (from 1 to ∞) converges.

Question 18

What is the root test?

Answer 18

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.

Question 19

what is the ratio test?

Answer 19

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.

Question 20

what is conditional convergence?

Answer 20

When a series is convergent, but not absolutely so.

Question 21

Define Power series and Radius of convergence.

Answer 21

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.

Question 22

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

Answer 22

(a) absolutely convergent

(b) divergent

Question 23

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?

Answer 23

R=C

Question 24

When is a FUNCTION bounded?

Answer 24

when its range is bounded

Question 25

define the limit of f at the point x

Answer 25

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|

Question 26

when is a function continuous at a point?

Answer 26

When the left limits and right limits at that point exist and coincide with f(x)

Question 27

If the sequence a(n) does not converge, what can be said of the corresponding series?

Answer 27

It also does not converge

Question 28

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?

Answer 28

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