The sum to infinity of a series depends on the type of series.
- For arithmetic series, the sum to infinity is undefined because the sum of terms leads to either positive or negative infinity.
- For geometric series, the sum to infinity is also undefined when the absolute value of the common ratio (|r|) is greater than or equal to 1. However, if |r| is less than 1, the sum to infinity of a geometric series can be calculated.
The sum to infinity of a geometric series with a common ratio less than 1 can be calculated using the following formula:
S∞ = a / (1 - r)
where:
- S∞ is the sum to infinity
- a is the first term
- r is the common ratio
For example, the sum to infinity of the geometric series 1 + 1/2 + 1/4 + 1/8 + ... is:
S∞ = 1 / (1 - 1/2) = 2
This means that as the number of terms in the series approaches infinity, the sum approaches 2.
