Geometric Series

Thus Without Loss Of Generality a geometric sequence can be written as : $a,ar,ar^2,ar^3,ar^4,\ldots\,$ where ''r'' ≠ 0 is the common ratio and ''a'' is a Scale Factor . Thus the common ratio gives a family of geometric sequences whose starting value is determined by the scale factor. Pedantically speaking, the case ''r'' = 0 ought to be excluded, since the common ratio is not even defined; but the sequence that is always 0 is included, by convention. FORMULA Progressions allow the use of a few simple formulae to find each term. The ''nth'' term can be defined as : $a_n = a\,r^{n-1} \quad \mbox{where n is an integer such that }n \ge 1$ The common ratio is then : $r=\left(rac{a_n}{a} ight)^rac{1}{n-1} \mbox{or } r=\sqrt$ {Link without Title} {rac{a_n}{a}} \quad \mbox{where n is an integer such that }n \ge 2 and the scale factor is : $a=rac{a_n}{r^{n-1}}.$ EXAMPLES A sequence with a common ratio of 2 and a scale factor of 1 is :: 1, 2, 4, 8, 16, 32, .... A sequence with a common ratio of 2/3 and a scale factor of 729 is :: 729 (1, 2/3, 4/9, 8/27, 16/81, 32/243, 64/729, ....) = 729, 486, 324, 216, 144, 96, 64, .... A sequence with a common ratio of −1 and a scale factor of 3 is ::3 (1, −1, 1, −1, 1, −1, 1, −1, 1, −1, ....) = 3, −3, 3, −3, 3, −3, 3, −3, 3, −3, .... This sequence's behaviour depends on the value of the common ratio. ::If the common ratio is: Negative, the terms will alternate between positive and negative. Greater than 1, there will be Exponential Growth towards Infinity (positive). Less than −1, there will be Exponential Growth towards Infinity (positive and negative). Between 1 and −1 but not zero, there will be Exponential Decay towards zero. Zero, the results will remain at zero. One, the progression is a constant sequence. −1, the progression is an alternating sequence (see Alternating Series ) A geometric progression with common ratio $otin \{0,\pm1\}$ shows Exponential Growth or Exponential Decay , as opposed to the Linear Growth (or decline) of an Arithmetic Progression such as 4, 15, 26, 37, 48, .... This result was taken by T.R. Malthus as the mathematical foundation of his ''Principle of Population''. Note that the two kinds of progression are related: taking the Logarithm of each term in a geometric progression yields an arithmetic one. GEOMETRIC SERIES A geometric series is the ''sum'' of the numbers in a geometric progression: : $\sum_{k=0}^{n} ar^k = ar^0+ar^1+ar^2+ar^3+\cdots+ar^n \,$ We can find a simpler formula for this sum by multiplying both sides of the above equation by $(1-r)$ , and we'll see that : $(1-r) \sum_{k=0}^{n} ar^k = a-ar^{n+1}\,$ since all the other terms cancel. Rearranging (for $r e1$ ) gives the convenient formula for a geometric series: : $\sum_{k=0}^{n} ar^k = rac{a(1-r^{n+1})}{1-r}$ Note: If one were to begin the sum not from 0, but from a higher term, say ''m'', then : $\sum_{k=m}^n ar^k=rac{a(r^m-r^{n+1})}{1-r}$ Differentiating the sum with respect to ''r'' allows us to arrive at formulae for sums of the form : $\sum_{k=0}^n k^s r^k$ For example: : $rac{d}{dr}\sum_{k=0}^nr^k = \sum_{k=0}^nkr^{k-1}= rac{1-r^{n+1}}{(1-r)^2}-rac{(n+1)r^n}{1-r}$ Infinite geometric series
	Both Are Valid Only For ''r'' < 1 This Last Formula Is Actually Valid In Every	"http://wwwinformationdelightinfo/encyclopedia/entry/Banach_algebra" class="copylinks">Banach Algebra , as long as the norm of ''r'' is less than one, and also in the field of ''p''-adic Numbers if ''r''<sub>''p''</sub> < 1 As in the case for a finite sum, we can differentiate to calculate formulae for related sums

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Geometric Series