BIOL 335 Lecture Notes - Lecture 15: Geometric Progression
STAT 443: Review of Geometric Series
A series of the form
a+ar +ar2+ar3+· · ·
is called a geometric series (or geometric progression). Each term in the sum is the previous
term multiplied by r.
The summation above is an infinite sum – there are an infinite number of terms being added
together. One might think that such a sum must therefore be ∞(or perhaps −∞ if all the
terms were negative), but in fact that is not necessarily the case. A geometric series of the form
above converges to a finite limit provided ris suitably small: if |r|<1 (i.e., −1< r < 1) then
the geometric series sums to a
(1 −r).
For other values of r, there is no finite limit possible in the infinite sum – in such cases we will
think of the summation as not being properly defined.
The sum of the first nterms can be shown to be
a+ar +ar2+· · · +arn−1=a(rn−1)
r−1
=a(1 −rn)
1−r
.
1
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
A series of the form a + ar + ar. 3 + is called a geometric series (or geometric progression). Each term in the sum is the previous term multiplied by r. The summation above is an in nite sum there are an in nite number of terms being added together. One might think that such a sum must therefore be (or perhaps if all the terms were negative), but in fact that is not necessarily the case. A geometric series of the form above converges to a nite limit provided r is suitably small: if |r| < 1 (i. e. , 1 < r < 1) then the geometric series sums to a (1 r) For other values of r, there is no nite limit possible in the in nite sum in such cases we will think of the summation as not being properly de ned.
