geometric sequenceGeometricSequence2004-09-26 05:14:442008-11-19 06:52:38Definitionsequencecommon ratio% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA sequence of the form
$$a,\,ar,\,ar^2,\,ar^3,\,\ldots$$
of real or complex numbers is called {\em geometric sequence}.\, \PMlinkescapetext{Characteristic} of the geometric sequence is thus that every two consecutive members of the sequence have the constant ratio $r$, called usually the {\em common ratio} of the sequence (if\, $ar = 0$, \PMlinkescapetext{strictly} speaking the ratio of members does not exist).
The $n^\mathrm{th}$ member of the geometric sequence has the \PMlinkescapetext{formula}
$$a_n = ar^{n-1}.$$
Let\, $a \neq 0$.\, The sequence is convergent for\, $|r| < 1$\, having the \PMlinkname{limit}{LimitOfRealNumberSequence} 0, and for\, $r = 1$\, having as constant sequence the limit $a$.
When the members of the sequence are positive numbers, each member is the geometric mean of the preceding and the following member; the name ``geometric sequence''(or ``geometric series'') is due to this fact (a \PMlinkescapetext{comparable} fact is true for the harmonic series and harmonic mean).