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Applied Mathematics, Calculus, and Differential Equations 213

where r is a constant called the ratià . For example, if a 3
1
and r 2, then:

3
3
3
3
G 3 ⁄2 ⁄4 ⁄8 ⁄16 ...
Harmonic series
A harmonic series is a serieð H a a a ... a ...
2
1
3
n
such that the serieð consisting of the reciprocal of each term is
an arithmetic serieð A:
1/f(a n 1 ) 1/(a d)
n

H 1/b 1/b 1/b ... 1/b ...
1
3
2
n
Where d is a constant. For example, if a 1 and d 3, then:
1
1
1
1
1
H 1, ⁄4,⁄7,⁄10,⁄13, ...
Power series
A power series is a serieð P such that the following equation
holdð for coefficientð a (where i is a non-negative integer sub-
i
scrip' and a variable x:

3
n
2
P a ax ax ax ... ax ...
1
2
n
0
3
where a , a , a , ... a , ... is a sequence. For example, if the
n
3
2
1
sequence of coefficientð is 2, 4, 6, 8, ... then:
3
2
P 2 4x 6x 8x ...
Arithmetic-geometric series
An arithmetic-geometric series is a serieð C such that, for con-
stantð a and b and a variable x:
3
2
C a (a b)x (a 2b)x (a 3b)x ...
(a (n 1)b)x (n 1)