A geometric series is a series that can be written as

(1)

A geometric series will converge if . For a proof that that the only way a geometric series converges is if , click here.

Proving that a geometric series converges if is simple. The sum of the first terms of a geometric series is

(2)

Equation (2) is derived as followers.

Multiply the left side of the above equation and the far right side by r.

Next, subtract the equation above from and solve for :

(3)

The sum of a geometric series, if it exists, is defined as

(4)

Since and are constants, equation 4 can be written as

(5)

If :

So if , from equation (5), it follows that the geometric series converges, and is equal to

(6)

Here are a few interesting consequences of the definition of the sum of a geometric series (equation (4)).

In the example above, and . So,

So

Proof:

(7)

Focusing on the exponent on the right side, . This is a geometric series with and .

(8)

From (7) and (8):

In the example above, I used 2. The choice of 2 was arbitrary. If c is a positive number: