Geometric series: Difference between revisions

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imported>Peter Schmitt
(include complex numbers)
imported>Peter Schmitt
(remove "minus")
Line 4: Line 4:
i.e., the quotient ''q'' of two consecutive terms is the same for each pair.
i.e., the quotient ''q'' of two consecutive terms is the same for each pair.


A geometric series converges if and only if &minus; |''q''|<1.
A geometric series converges if and only if |''q''|<1.


Its sum is <math> a \over 1-q </math> where ''a'' is the first term of the series.
Its sum is <math> a \over 1-q </math> where ''a'' is the first term of the series.

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A geometric series is a series associated with an infinite geometric sequence, i.e., the quotient q of two consecutive terms is the same for each pair.

A geometric series converges if and only if |q|<1.

Its sum is where a is the first term of the series.

Power series

Any geometric series

can be written as

where

The partial sums of the power series are

because

Since

there is

and the geometric series converges for |x|<1 with the sum

and diverges for |x| ≥ 1.