Appendix E: Sigma Notation
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Definition: Sequence
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A sequence is a function a(n) (written an) who’s domain is
the set of natural numbers {1, 2, 3, 4, 5, ….}. an is called
the general term of the sequence.
The output of a sequence can be written as {a1, a2, a3, …,
an-1, an, an+1, …}, where an is a term in a sequence, an-1 is
the term before it, and an+1 is the term after it.
Sequences can be either finite (their domains are {1, 2, 3,
…, n}) or infinite (their domains are {1, 2, 3, ….}).
A sequence who’s input for the next term in the sequence
is the value of the previous term is called a recursive
sequence.
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Definition: Arithmetic Sequence
An arithmetic sequence is a sequence generated
by adding a real number (called the common
difference, d) to the previous term to get the next
term. The general term of an arithmetic is given
by an = a1 + d(n – 1) where a1 and d are any real
numbers.
Example Find the general term of the 7/3, 8/3, 3,
10/3, ….
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Definition: Geometric Sequence
A geometric sequence is a sequence generated by
multiplying the previous term by a real number (called
the common ratio r). The general term of a geometric
sequence is given by an = a1 r(n – 1) where a1 and r are
any real numbers, is called an geometric sequence.
Example Find the general term sequence 2, 2/5, 2/25,
2/125, …
TI: seq(ax , x, i start, i stop)
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Definition: Series
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A finite series is the sum of a finite number of
terms of a sequence.
An infinite series is the sum of an infinite
number of terms of a sequence.
We use sigma notation to denote a series. The
series does not have to start at i = 1, but i must
be in the domain of ai.
n
a
i 1
i
 a1  a2  ...  an
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Definition: Geometric Sequence
The nth partial sum is the sum of the first n terms
of a sequence. It MUST start at i = 1 with partial
sum notation. n
Sn   ai  a1  a2  ...  an
i 1
An infinite sum is the sum of all the terms of an
infinite sequence.

S   ai  a1  a2  ...
i 1
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Definition: Example
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1. Evaluate
.
i 2 i  1
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2. Evaluate
3 k
2
 .
k 2
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3. Evaluate  x k .
k 0
TI: sum(seq(ax , x, i start, i stop))
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Definition: Example
4. Write the sum in sigma notation.
3 4 5 6 7
   
5 7 9 11 13
5. Write the sum in sigma notation.
1+3+5+7+...+(2n-1)
6. Write the sum in sigma notation.
1  x  x  x  ...  (1) x
2
3
n
n
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Definition: Series
Sn 
n
 a1  an 
2

For a finite arithmetic series,

For an infinite arithmetic series, S  DNE


For a finite geometric series, Sn 
For an infinite geometric series,
if | r | < 1. It DNE otherwise.
a1 1  r n 
1 r
a1
S 
1 r
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Definition: Example
7. Find the 97th partial sum of an  2n  3.

i
1
8. Evaluate  2   .
i 1  3 
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Definition: Series Formulas
Let c be a constant and n a positive integer.
n
a.
 ca
i
i 1
n
 c  ai
i 1
n
b.
n
i
i
n
1  n
i 1
n
d.
n
 a  b   a  b
i 1
c.
n2 n
e.  i  
2 2
i 1
n
 c  cn
i 1
i 1
i
i 1
i
3
2
n
n
n
2
f . i   
3 2 6
i 1
n
 n  n  1 
g.  i  

2
i 1


n
2
3
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Definition: Series Formulas
n
9. Write a formula for the series in terms of n:
  i  3
2
i 1
10. If the interval [a, b] is split into n equal subintervals,
write a sequence xi that represents the x coordinate of the
left side, midpoint, and right side of each subinterval.
1 i
 3
11. Show that lim    1 
n 
 2
i 1 n  n
n
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