A geometric series is the sum of a geometric sequence with an infinite number of terms. Briefly, a geometric sequence is a type of sequence in which each subsequent term after the first term is determined by multiplying the previous term by a constant (not equal to 1). This constant is referred to as the common ratio. For example, the following is an infinite geometric sequence in which the common ratio (often denoted with the variable r) is r = ⅓ and the first term a1 = ⅓:
The general form of a geometric sequence is
where a1 is the first term and r is the common ratio. To determine any given term in the sequence, the following formula can be used:
As mentioned, a geometric series is the sum of an infinite geometric sequence. Referencing the above example, the partial sum of the first 6 terms in the infinite geometric sequence (or the partial geometric series) can be denoted and computed as follows:
The partial sum, Sn, of any given geometric series can be computed without the need to manually add each term by using the following formula:
For example, referencing the same example:
This formula only works when n is a finite number. In cases where n goes to infinity, a different formula must be used.
Infinite geometric series
An infinite geometric series is one in which n goes to infinity. In most cases we will be dealing with infinite geometric series, so from this point forward, the term "geometric series" references infinite geometric series. In cases where -1 < r < 1, the geometric series converges since , and the sum of the geometric series can be computed using the following formula:
In cases where |r| ≥ 1, the geometric series diverges (does not exist or approaches ±∞), and therefore has no sum.
Find the sum of the following geometric series:
i. a1 = 1 and r = ½, so:
ii. a1 = 2 and r = ⅔, so:
iii. Since r = 3 > 1, the sum of the series approaches infinity, and therefore diverges, so we cannot compute its sum. It is worth noting that a divergent series does not necessarily have to be ±∞. If the series oscillates, it does not approach one particular value, and the sum of its terms will differ depending on n.
Applications of geometric series
In addition to mathematics, geometric series are used in various fields of study, such as science and business.
Calculating the exact value of a repeating decimal
Repeating decimals can be represented as a geometric series, and thereby be expressed in the form of a fraction. For example, the repeating decimal 0.3 can be expressed as a geometric series with a1 = 0.3 and r = 0.1 such that the first 4 terms of the series are:
0.3 + 0.03 + 0.003 + 0.0003 + ...
Since the series converges, the sum can be computed as follows:
Find the exact value of 0.1523.
The repeating decimal can be written as 0.1523 + 0.00001523 + 0.000000001523 + ..., which is a geometric series with a1 = 0.1523 and r = 0.0001. The exact value can be computed as follows:
Modeling exponential growth or decay
Geometric series can be used to model exponential growth or decay. Consider an experiment involving bacterial growth in a petri dish. Given that there are 10,000 bacterial cells in the petri dish, and the rate at which the bacteria grow is halved with each passing hour, the exponential decay of the growth of the bacteria can be modeled with the following function:
where t is the number of hours. Thus:
Note that the above is a geometric series in which a1 = 10000 and r = ½. The partial sum, St of the number of bacterial cells after t hours can therefore be represented as a geometric series. This allows us to determine the maximum number of bacteria, S, in the petri dish as t→∞:
Geometric series in business
Geometric series can also be used to determine the value in an account over a period of time as a result of compound interest. Suppose that you contribute $1000 a year for 20 years to an account that compounds annually at a rate of 5%. The amount in the account after 20 years can be computed using the formula:
where t is time in years. Thus, the first contribution will earn interest for 20 years, the second will earn interest for 19 years, the third will earn interest for 18 years, and so on, such that:
The above is a geometric sequence such that a1 = 1050 and r = 1.05. The partial sum of the series can therefore be computed for the first 20 terms as:
Thus, the account total after 20 years is $34,719.25.