Sequences

Question 1

Definition of Convergence

Answer 1

For every epsilon>0 there exists a natural number N such that n is a natural number, n>=N implies that |s_n-s|→s

Question 2

Let s_n and a_n be sequences and let s be a real number, if for some k>0, m in the natural numbers and a_n→0

Answer 2

|s_n-s|<k>n| for all n&gt;=m, then s_n→s</k>

Find an upper bound for numerator and lower for denominator

Question 3

Theorem 4.2.1 (Basic Properties of Limits)

Answer 3

a) lim (s_n + t_n) = s + t
b) lim (ks_n) = ks and lim (k + s_n) = k + s
c) lim (s_nt_n) = st
d) lim (s_n/t_n) = s/t if t_n nor t =0

Question 4

Theorem 4.2.4 Conservation of Order Relation

Answer 4

Suppose s_n → s and t_n → t. If s_n _<_t_n for all n in N, then

s _<_t

Question 5

Thm 2 or 4.2.5 Corollary

If t_n→t and t_n_>_0

Answer 5

If t_n→t and t_n_>0 for all n, then t>_0

Question 6

Thm 3 or 4.2.7 Theorem

(Sequence of Ratios)

Answer 6

Let (s_n) be a sequence of positive numbers. If (s_n+1/s_n)→ L and L<1, then lim s_n = 0

Question 7

Definition of Infinite Limits

A sequence s_n is said to diverge to +infinity (s_n →+oo)

Answer 7

if to each M in R there is a positive integer N such that s_n>M for all n>N

Question 8

Definition of Infinite Limits

A sequence s_n is said to diverge to -infinity (s_n →-oo)

Answer 8

if to each M in R there is a positive integer N such that s_n < M for all n>N

Question 9

Thm 4.2.12

Suppose s_n and t_n are sequences such that s_n_<_t_n for all n in N

Answer 9

If lim s_n = + infinity, then lim t_n = + infinity

If lim t_n = - infinity, then lim s_n = - infinity

Question 10

Thm 4.2.13 Inverse of Infinite Limits

Let sn be a sequence of positive numbers

Answer 10

Then lim s_n = + infinity if and only if lim 1/s_n = 0

Question 11

Increasing if s_n_<s_n+1 and decreasing if s_n>_s_n+1

A sequence is monotone if

Answer 11

it is either increasing or decreasing

Question 12

Monotone Convergence Theorem

Answer 12

A monotone sequence is covergent if and only if it is bounded

Question 13

Thm 4.3.8 Unbounded Monotone Sequences

Answer 13

If s_n is unbnd increasing, then s_n→ + infinity

If s_n is unbnd decreasing then s_n→ - infinity

Question 14

Thm 4.3.12 Cauchy Convergence Criterion

Answer 14

A sequence of real numbers is convergent iff it s a Cauchy sequence

(Don’t forget every convergent sequence is bounded!)

Question 15

Definition of Subsequences

Given s_n, let n_k be a sequence of positive integers such that

Answer 15

n₁<n>2<n>3&lt;...</n></n>

That is, n_k is an increasing sequence s_nk is a subsequence of s_n

Question 16

If a sequence converges to a real number s,

Answer 16

then every subsequence of s_n also converges to s

Question 17

If a sequence has two subsequences that converge to different numbers,

Answer 17

then the sequence DOES NOT converge

Question 18

Bolzano-Weierstrass (bounded sets have an accumulation pt) for Sequences

Answer 18

Every bounded sequence has a convergent subsequence