1. Arithmetic and Geometric Series
April 13, 2007

2. Sequence vs. Series
Sequence – a set of numbers that follow a general rule (nth term formula)
Examples: 1, 8, 15, 22, 29, …
-8, 2, -1/2, 1/8, …
Series – the sum of the terms in a sequence of numbers
Examples: …
– 1/2 + 1/8 - …

3. Finite vs. Infinite a Finite series has an end
Example: -9 – …
an Infinite series continues indefinitely
Example: …

4. Finite Arithmetic Series
WARNING: you MUST have an Arithmetic Series to use this formula
Reminder: the nth term formula for an arithmetic series is
an = a1 + (n – 1)d

5. Example
Find the sum of the first 60 terms in the arithmetic series …
n = a1 = a60 = ?
a60 = a1 + (60 – 1)d
= 9 + (60 – 1)5
= 304
S60 = 60 ( ) 2
= 30 (313)
= 9390

6. Finite Geometric Series
WARNING: you MUST have a Geometric Series to use this formula

7. Example Sn = a1 (1 – rn) – r
Find the sum of the first ten terms of the geometric series
16 – – …
n = 10 a1 = 16 r = -3
S10 = 16 [1 – (-3)10]
1 – (-3)
= 16 (1 – 59,049) 4
= -236,192

8. Infinite Geometric Series (condition |r|<1)
Sn = a1 (1 – rn)
1 – r
if |r|<1, what happens to rn when n becomes very large (i.e. approaches infinity)
rn  0
Infinite Geometric Series
S = a1__

9. Example S = a1_ 1 – r Find the sum of the series 21 – 3 + 3/7 - …
Is |r| < 1?
r = -1/7, so |r| < 1
S = ____
1 – (-1/7)
S = 147 8
Fancy Math Language: Converges to 147/8

10. Example S = a1_ 1 – r Find the sum of the series 1 + 3 + 9 + 27 + …
Is |r| < 1?
r = 3, so |r| > 1
the condition is not met
So the sum does not exist
Fancy Math Language: Diverges