Summary Sequences and series notes for exam with examples

SEQUENCES AND SERIES – MOST IMPORTANT INFORMATION

SEQUENCES
– a rule assigning to each natural number n a unique real number xn

Examples
- Constant sequence: (1,1,1,…)
- Fibonacci sequence (0,1,1,2,3,5, … , xn+2 + xn-1)
- (1/n) = (1,1/2,1/3,…)

CONERGENCE THEOREM
A sequence xn converges to the limit l if and only If for all ε > 0 there exists a N such that
|xn – l| < ε

A sequence is convergent if it converges to some limit l.
A sequence which is not convergent is divergent.

EXAMPLE 1
Prove that a sequence converges to 0 using Convergence Theorem
1/n -> 0 as n -> ∝

|1/n – 0| < ε
n > 1/ε
then for all n>N we have n > N ≥ 1/ε and therefore |1/n – 0| < ε as required.


EXAMPLE 2
Prove that a sequence does not converge to a limit
xn = ((-1)n) does not converge to 1

Take ε = 1 then for all n>N we must have |xn – 1| < 1
We note that

|xn – 1| = 2 if n is odd and |xn – 1| = 0 if n is even, then
For any proposed N(1) we can always find an odd n > N(1) and |xn – 1 | = 2 ≮ 1

UNIQUNESS OF LIMIT THEOREM
A sequence can have at most one limit.

BOUNDEDNESS THEOREM
1. We say that a sequence xn is bounded above if ∃H ∈ R such that xn ≤ H for all n. For
example: xn = -n
2. We say that a sequence is bounded below if ∃h ∈ R such that xn ≥ h for all n. For
example: xn = n or xn = n2
3. A sequence is bounded if it’s bounded above and bounded below. For example:
xn = ((-1)n)

, - Any convergent sequence is bounded, but not every bounded sequence is
convergent (for example: xn = ((-1)n) it is bounded but divergent: oscillates between 1
and -1)
- Any unbounded sequence is divergent.
Examples: xn = 2n or xn = n

COMBINATION THEOREM
- For any α ∈ R we have αxn → αl as n →∝
- xn + yn → l + m as n →∝
- xn yn → lm as n →∝
- if m ≠ 0 then xn/yn → l/m as n →∝

EXAMPLE 3
(3n + 1)/(n + 2) → 3 as n →∝

Rewrite (3n + 1)/ (n + 2) = (3 + 1/n) / (1+ 2/n)

1/n → 0 as n →∝
2 x 1/n = 2 x 0 → 0 as n →∝

So (3 + 1/n) / (1+ 2/n) = 3/1= 3 as n →∝

SANDWICH RULE
Suppose yn → l as n →∝ and zn → l as n →∝ and yn ≤ xn ≤ zn
Then xn → l as n →∝

EXAMPLE 4
sin(n) / 4n3 → 0 as n →∝

-1 ≤ sin(n) ≤ 1
-1/4n3 ≤ sin(n) / 4n3 ≤ -1/4n3

-1/4n3→ 0 as n →∝
1/4n3 → 0 as n →∝

Using Sandwich Rule: sin(n) / 4n3 → 0 as n →∝

MONOTONIC SEQUENCES
1. A sequence xn is increasing if xn+1 ≥ xn ∀n (strictly increasing if xn+1 > xn)
2. A sequence xn is decreasing if xn+1 ≤ xn ∀n (strictly decreasing if xn+1 < xn)

MONOTONIC CONVERGENCE THEOREM
1. If a sequence is decreasing and bounded below, it’s convergent (converges to
infimum).
2. If a sequence is increasing and bounded above, it’s convergent (converges to
supremum).