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Limit of Geometric Serries

Proposition: For q in R, |q| < 1, [(sum_{0 <= i <= n} qⁱ)]_n converges to 1/1-q:

forall q in R: |q| < 1 => (sum_i=0^oo qⁱ) = 1/1-q.

Proof: Take arbitrary q in R with |q| < 1. We then have

(sum_i=0^oo qⁱ) =

lim_{n -> oo} (sum_{0 <= i <= n} T) =

lim_{n -> oo} qⁿ-1/q-1 =

lim_{n -> oo} (qⁿ-1)/lim_{n -> oo} (q-1) =

(lim_{n -> oo} qⁿ) - (lim_{n -> oo} 1)/q-1 = (*)

0-1/q-1 =

1/1-q.

(*) The fact lim_{n -> oo} qⁿ = 0 has to be shown in a separate proof.

Author: Wolfgang Schreiner
Last Modification: December 14, 1999

(sum_i=0^oo `q`ⁱ)	=
lim_{n -> oo} (sum_{0 <= i <= n} `T`)	=
lim_{n -> oo} `q`ⁿ-1/`q`-1	=
lim_{n -> oo} (`q`ⁿ-1)/lim_{n -> oo} (`q`-1)	=
(lim_{n -> oo} `q`ⁿ) - (lim_{n -> oo} 1)/`q`-1	=	(*)
0-1/`q`-1	=
1/1-`q`.