Chu, Inversion techniques and combinatorial identities--strange evaluations of hyper geometric series
, Pure Math.
Since Motomura (1932) developed the geometric series
model to describe the structure of an aquatic community, ecologists have built many models to fit species-abundance data derived from communities or collections.
It covers functions and change, rate of change; the derivative, short-cuts to differentiation, using the derivative, accumulated change; the definite integral, anti-derivatives and applications, probability, functions of several variables, mathematical modeling using differential equations, and geometric series
. First published in 1999 and updated here from the 2010 edition.
Note that [summation].sup.n-1.sub.k=0] (r[[omega].sup.-m]) is a geometric series
with first term 1 and common ratio r[[omega].sup.-m] Using the formula for the sum of a geometric series
, we have
With M0 and MC it can be assumed that cosmic thermal energy density, matter energy density and the critical energy density are in geometric series
and the geometric ratio is 1 + 1n ([M.sub.0/[M.sub.C]).
One can use intuition alone or calculus and polar co-ordinates, but herein we offer a solution we have not seen presented elsewhere using the sum of a geometric series
applied in an unexpected way.
The addition of a Trapezoid (TRPC) and Radius (RDIC) Shape complete the compact Architectural Sconce Geometric Series
, and the LMC-30 (pictured) delivers a new powerful addition to the popular Laredo Series.
No model has a low distribution test statistic ([r.sub.test] < 10) and only the Geometric series
model (Geoser) had a low octave test statistic ([oc.sub.test] =7.57).
This is easily proven by using an infinite geometric series
Zeilberger, Hyper geometric series
acceleration via the WZ method, Electron.
Blank Frank (all works 2009), for example, is a simple arrangement of two lightly painted kidney or lung shapes (an umbilical appendage growing from one) hovering over an inverted dark blue mound; floating to the left of these forms, a more geometric series
of brushy lines punctuates the otherwise anemic field.
Nowadays, the geometric series
play a key role in digital number system, for example, converting a base-2 number into a base-10 number.