Geometric sequence - Geometric series

A geometric sequence is obtained by multiplying each term by a constant "a" :
Un+1 = aUn
"a" is named the "ratio" of the geometric sequence.
For instance starting from 5 with ratio 2 : 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, ...

The n+1th term is given by

Un = U0 × an
In previous example, the 10th term is 5x29=5x512=2560.

The sum Sn of the n+1 first terms ("geometric series") is obtained by calculating Sn-aSn :
  U0+ U1 + U2+....+Un-2+ Un-1+ Un
        -aU0-aU1-....-aUn-3-aUn-2-aUn-1-aUn
=Sn(1-a)=U0-aUn=U0(1-an)

Sn = U0(1-an)/(1-a)

Application : sum of powers of 3 :
It is a geometric series with ratio 3, the sum is Sn = (3n-1)/2.
for instance : 1+3+9+27+81+243+729+2187+6561+19683=(310-1)/2=(59049-1)/2=29524

Convergence

If |a|<1, Un+1<Un. The terms decrease towards 0. We may interrest in the convergence of the series, that is the sequence of Sn when n grows to infinity.
If |a|<1, |an| approaches 0 and Sn approaches

S = U0/(1-a)

For instance 1+1/2+1/4+1/8+...+1/2n approaches 1/(1-1/2)=2

 

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