Saturday, April 2, 2011

Geometric Series

  • GS.ID.1: Geometric Series Formula
Proof



Multiply s with r , therefore


Subtract rs from s , therefore


therefore,

As n goes to infinity, the absolute value of r must be less than one for the series to converge, therefore ,



  • GS.ID.2: Geometric Series of Exponential

As n goes to infinity, the absolute value of e converges





  • GS.ID.3: Geometric Series Derivatives
Given

(1)

First derivative

(2)
Second derivative

(3)
(4)

(5)
from (3) and (5)

(6)


from (2) and (6) 
 
(7)
 
 (8)
 

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