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Summary Sequences and series notes for exam with examples

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Module
Mathematics

Institution
City University (City)

Notes for exam from sequences and series (includes sequences, series and power series) with examples to understand the theorems. All of the important theorems have been included.

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Uploaded on January 27, 2022
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Written in 2021/2022
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sequences and series
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sequences and series examples

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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).

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