1 sequences and summations elementary discrete mathematics jim skon
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1
Sequences and Summations
Elementary Discrete Mathematics
Jim Skon
Sequences and Summations 2
Sequences
Consider the function ƒ: N+ R where
ƒ(x) = 1/xThen:
The image of this function can be said to form a sequence:
f(1)1, f(2)1
2, f(3)
1
3, f(4)
1
4, ...
1,1
2,1
3,1
4,1
5,1
6,...
Sequences and Summations 3
Sequences
Sequence: A sequence is a function from a subset of the set of integers to a set S.
The image of the nth integer is denoted by an.
an is called a term of the sequence.
Then, for the function above, we can define the sequence as:
an , where an 1
n
Sequences and Summations 4
Sequences
The terms of this the previous sequence are:
a1, a2, a3, a4, a5, a6,...Which is:
Where:
,...6
1,
5
1,
4
1,
3
1,
2
1,1
an , where an 1
n
Sequences and Summations 5
Sequences
Again, if the sequence is defined as:
Then11 a
an , where an 1
n
2
12 a
3
13 a
4
14 a
5
15 a
6
16 a
7
17 a
and so on...
Sequences and Summations 6
Sequence Examples
nnn bb 1 where,
nnn cc 5 where,
nnnn aa 532 where,
Sequences and Summations 7
Summation NotationSymbolic way to define the sum of a series.Example:
i - index of summation
1 - lower limit of summation
100 - upper limit of summation
100 . . . 3 2 1 100
1
i
i
Sequences and Summations 8
Summation Notation Examples
6
2
3
i
i
5
1 )1(
1
k kk
m
mj
mj
Sequences and Summations 9
Summation Notation Examples
4
1
3
1i j
ij
Ss
sf )(
n
i
m
j j
i
1 12
Where S = {1, 2, 3, 4, 5}f:RRf(x) = 3x2+1