How do you evaluate a geometric series?

You evaluate a geometric series differently depending on whether it's finite or infinite.

Evaluating a Finite Geometric Series

A finite geometric series has a specific number of terms. The formula to evaluate it is:

S_n = a₁ * ((1 - rⁿ) / (1 - r))

Where:

• S_n is the sum of the first n terms of the series.
• a₁ is the first term of the series.
• r is the common ratio (the value you multiply each term by to get the next term).
• n is the number of terms in the series.

Example:

Consider the series: 2 + 6 + 18 + 54

• a₁ = 2
• r = 3 (6/2 = 3, 18/6 = 3, and so on)
• n = 4

Therefore, S₄ = 2 ((1 - 3⁴) / (1 - 3)) = 2 ((1 - 81) / (-2)) = 2 (-80 / -2) = 2 40 = 80

So, 2 + 6 + 18 + 54 = 80.

Evaluating an Infinite Geometric Series

An infinite geometric series continues indefinitely. The sum of an infinite geometric series converges (approaches a finite value) only if the absolute value of the common ratio, |r|, is less than 1 (i.e., -1 < r < 1). If |r| ≥ 1, the series diverges and does not have a finite sum.

The formula for the sum of a convergent infinite geometric series is:

S = a₁ / (1 - r)

Where:

• S is the sum of the infinite series.
• a₁ is the first term of the series.
• r is the common ratio (and -1 < r < 1).

Example:

Consider the series: 1 + 1/2 + 1/4 + 1/8 + ...

• a₁ = 1
• r = 1/2

Therefore, S = 1 / (1 - 1/2) = 1 / (1/2) = 2

So, 1 + 1/2 + 1/4 + 1/8 + ... = 2.

In summary, to evaluate a geometric series, identify whether it's finite or infinite, determine the first term (a₁) and the common ratio (r), and then apply the appropriate formula. Remember that an infinite geometric series only has a finite sum if |r| < 1.