This will be needed in an geometric series with ratio -4.

Now life gets tricky: these numbers are in a simple geometric series.

His silver work had smooth surfaces and was based on the geometric series.

The terms on the left make a geometric series, and we know that converges.

The first equality is given by the formula for a geometric series in each term of the product.

The sum of a geometric series is itself a result even older than Euler.

The following table shows several geometric series with different common ratios:

Above we have used the standard mathematical formula for the sum of a geometric series.

The same strategy works for any finite geometric series.

In general, an Engel expansion with constant terms is a geometric series.

Using the geometric series, we then find (for finite )

He expressed the solution to the problem as an infinite geometric series, whose sum was 4/3.

The sequence 1 can be thought of as a geometric series with the ratio 1.

It is possible to calculate the sums of some non-obvious geometric series.

Archimedes succeeded in summing what is now called a geometric series.

To derive this formula, first write a general geometric series as:

I. Motomura developed the geometric series model based on benthic community data in a lake.

Using an ingenious trick, he was able to reduce this evaluation to the sum of geometric series.

An approximate idea of the Householder's method derives from the geometric series.

Grandi's series is just one example of a divergent geometric series.

The series is a geometric series whose sum can be expressed analytically.

A geometric series is the sum of the numbers in a geometric progression.

The summation formula for geometric series remains valid even when the common ratio is a complex number.

On the other hand, R(z) expanded as a geometric series gives:

Alternatively, one can start with the finite geometric series (t 1)