Arithmetic & Geometric Series

Series, like sequences, come in both arithmetic and geometric varieties.

• Arithmetic Series: Of the form

  
  
  

• Geometric Series: Of the form

  

Thus, the series:

2 + 5 + 8 + 11 + ...

is arithmetic with  a₀ = 2  and  r = 3 . The series:

2 + 1 + 1/2 + 1/4 + 1/8 + ...

is geometric with  a₀ = 2  and  r = 1/2 .

It isn't difficult to calculate the respective partial sums. For arithmetic series:

We proceed from the third to the fourth line above by way of a useful formula for the sum of the first  N  positive integers.

  Summing 1 + 2 + 3 + ... + N

The sequence of partial sums for an arithmetic series grows without bound whenever  r > 0  and decreases without bound whenever  r < 0 . The partial sums "settle down" only in the trivial case where both  a₀ = 0  and  r = 0 . As a result, all non-trivial arithmetic series diverge, and we do not speak of their "sum".

Geometric series are another matter. Notice that:

So:

Taking the difference gives us:

From this we may solve for the partial sum:

In this partial sum, the only term dependent on  N  is  r ^{N + 1} . When  |r| < 1 , this term is an exponential that decays to  0  as  N  increases. In this case, the series will converge to the sum: