introduction to sequences. the real number system/sequences of real numbers/introduction to...
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Introduction to Sequences
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
Sequences
DefinitionDefinition
A sequence (an)=(a1, a2, a3, …) is a rule that assigns number an to every positive integer n.
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
Examples of Sequences
SEQUENCES
(3., 3.1, 3.14, 3.141, 3.1415,…).
n −1n
, n =1, 2, 3,K⎛
⎝⎜⎞
⎠⎟= 0,
12,23,K
⎛
⎝⎜⎞
⎠⎟,
(n) = (1,2,3,…),
(2n-1) = (1,3,5,…),
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
SEQUENCES
Bar codes are finite sequences.
Examples of Sequences
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
Examples
VIBRATING BODIES
A string of a piano vibrates at a frequency
determined by its length.
The Fundamental Frequency or The
Fundamental Tone.
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
Examples
OVERTONES
Along with its fundamental
frequency, the string vibrates also at higher frequencies producing overtones, also known
as harmonics.
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
SEQUENCES OF FREQUENCIES
Sounds produced by vibrating bodies consist always sequences of
frequencies: the fundamental frequency
together with the overtones.
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
SEQUENCES OF FREQUENCIES
The length of a piano string is determined by
the sequence of frequencies it produces.
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
SEQUENCES OF FREQUENCIES
“Can you hear the shape of a drum?”
one cannot hear the shape of a drum.
Answer
In general no,
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
Definition
OPERATIONS ON SEQUENCES
Let (an) and (bn) be sequences and k ∈ ℝ. Sum of Sequences: (an) + (bn) = (an + bn)
Product of a number and a Sequence:
k (an) = (kan).
Product of Sequences: (an)∙(bn) = (an∙bn).
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
Problem
OPERATIONS ON SEQUENCES
Let
(an) = (0, -2, 4, -6,…)
and (bn) = (-2, -4, -6, …).
Compute the general term cn of the sequence (cn) = (an) + (bn).
Find formulae for the terms an and bn.
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
OPERATIONS ON SEQUENCES
There are many possible formulae. Look for the “simplest” formula.
Problem
Let
(an) = (0, -2, 4, -6,…)
and (bn) = (-2, -4, -6, …).
Find formulae for the terms an and bn.
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
OPERATIONS ON SEQUENCES
Let
(an) = (0, -2, 4, -6,…)
and (bn) = (-2, -4, -6, …).
Find formulae for the terms an and bn.
Problem
Solution an = (-1)n+1 2(n - 1)
bn = - 2n
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
OPERATIONS ON SEQUENCES
Let
(an) = (0, -2, 4, -6,…)
and (bn) = (-2, -4, -6, …).
Problem
Compute the general term cn of the sequence (cn) = (an) + (bn).
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
OPERATIONS ON SEQUENCES
Solution
an = (-1)n+1 2(n - 1)
bn = -2n
cn = (-1)n+1 2(n-1) - 2n
c
n= −2, if n odd
2 −4n, if n even
⎧⎨⎪
⎩⎪
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
Sequences
A sequence (an)=(a1, a2, a3, …) is a rule that assigns number an to every positive integer n.
The Real Number System/Sequences of Real Numbers/Introduction to Sequences by Mika Seppälä
TONES AS SEQUENCES