Geometric series

A geometric series is a sum of terms in which two successive terms always have the same ratio. For example,

4 + 8 + 16 + 32 + 64 + 128 + 256

is a geometric series with common ratio 2.

The sum of a geometric series can be computed quickly with the formula

       n        xn+1 - xm
       ∑ xk  =  ---------
      k=m        x - 1

which is valid for all natural numbers m ≤ n and all numbers x≠ 1 (or more generally, for all elements x in a ring such that x - 1 is invertible). This formula can be verified by multiplying both sides with x - 1 and simplifying.

Using the formula, we can determine the above sum: (2⁹ - 2²)/(2 - 1) = 508. The formula is also extremely useful in calculating annuities: suppose you put $2,000 in the bank every year, and the money earns interest at an annual rate of 5%. How much money do you have after 6 years?

2,000 · 1.05⁶ + 2,000 · 1.05⁵ + 2,000 · 1.05⁴ + 2,000 · 1.05³ + 2,000 · 1.05² + 2,000 · 1.05¹

= 2,000 · (1.05⁷ - 1.05)/(1.05 - 1)

= 14,284.02

An infinite geometric series is an infinite series whose successive terms have a common ratio. Such a series converges if and only if the absolute value of the common ratio is less than one; its value can then be computed with the formula

      ∞          1
      ∑ xk  =  ------
     k=0       1 - x

which is valid whenever |x| < 1; it is a consequence of the above formula for finite geometric series by taking the limit for n→∞.

This last formula is actually valid in every Banach algebra, as long as the norm of x is less than one, and also in the field of p-adic numbers if |x|_p < 1.