Geometric series

A geometric series issumtermswhich two successive terms always havesame ratio. For example,

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

isgeometric seriescommon ratio 2. This issame as 2 * 2^x where xincreasing by oneeach number. Itcalledgeometric series becauseoccurs when comparinglength, area, volume, etc. ofshapedifferent dimensions.

The sum ofgeometric series can be computed quickly withformula

whichvalidall natural numbers m ≤ nall numbers x≠ 1 (or more generally,all elements x inring such that x - 1invertible). This formula can be verified by multiplying both sidesx - 1simplifying.

Usingformula, we can determineabove sum: (2⁹ - 2²)/(2 - 1) = 508. The formulaalso extremely usefulcalculating annuities: suppose you put $2,000 inbank every year, andmoney earns interest at an annual rate5%. 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 seriesan infinite series whose successive terms havecommon ratio. Suchseries converges ifonly ifabsolute value ofcommon ratioless than one; its value can then be computed withformula

whichvalid whenever |x| < 1;isconsequence ofabove formulafinite geometric series by takinglimitn→∞.

This last formulaactually validevery Banach algebra, asasnormxless than one,also infieldp-adic numbers if |x|_p < 1.

Also usefulmention:

which can be seen as x timesderivative ofinfinite geometric series. This formula only works|x| < 1, as well.

See also: infinite series