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The geometric series on the real line. In mathematics, the infinite series 1 / 2 + 1 / 4 + 1 / 8 + 1 / 16 + ··· is an elementary example of a geometric series that converges absolutely. The sum of the series is 1. In summation notation, this may be expressed as
The convergence of a geometric series can be described depending on the value of a common ratio, see § Convergence of the series and its proof. Grandi's series is an example of a divergent series that can be expressed as + +, where the initial term is and the common ratio is ; this is because it has three different values.
It is useful to figure out which summation methods produce the geometric series formula for which common ratios. One application for this information is the so-called Borel-Okada principle: If a regular summation method assigns = to / for all in a subset of the complex plane, given certain restrictions on , then the method also gives the analytic continuation of any other function () = = on ...
For some other divergent geometric series, including Grandi's series with ratio −1, and the series 1 + 2 + 4 + 8 + ⋯ with ratio 2, one can use the general solution for the sum of a geometric series with base 1 and ratio , obtaining , but this summation method fails for 1 + 1 + 1 + 1 + ⋯, producing a division by zero.
The addition of two divergent series may yield a convergent series: for instance, the addition of a divergent series with a series of its terms times will yield a series of all zeros that converges to zero. However, for any two series where one converges and the other diverges, the result of their addition diverges.
is used for the series, and, if it is convergent, to its sum. This convention is similar to that which is used for addition: a + b denotes the operation of adding a and b as well as the result of this addition, which is called the sum of a and b. Any series that is not convergent is said to be divergent or to diverge.
In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σa n, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original ...
The more general class of p-series, =, exemplifies the possible results of the test: If p ≤ 0, then the nth-term test identifies the series as divergent. If 0 < p ≤ 1, then the nth-term test is inconclusive, but the series is divergent by the integral test for convergence.