Geometric series: Difference between revisions

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imported>Peter Schmitt
(→‎Power series: two remarks added)
imported>Peter Schmitt
(→‎Power series: replace "definitely")
Line 47: Line 47:
: <math> \sum_{k=1}^\infty a_k = { a \over 1-q }</math>
: <math> \sum_{k=1}^\infty a_k = { a \over 1-q }</math>
and diverges for |''x''| &ge; 1.  
and diverges for |''x''| &ge; 1.  
(It diverges definitely &mdash; to +∞ or &minus;∞ depending on the sign of ''a'' &mdash; for ''x''&ge;1.)
(The limit is +∞ or &minus;∞ depending on the sign of ''a'' for ''x''&ge;1.)

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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.

Then its sum is where a is the first term of the series.

Example

The series

is a geometric series with quotient

and first term

and therefore its sum is

Power series

Any geometric series

can be written as

where

The partial sums of the power series Σxk are

because

Since

it is

and the geometric series converges (more precisely: converges absolutely) for |x|<1 with the sum

and diverges for |x| ≥ 1. (The limit is +∞ or −∞ depending on the sign of a for x≥1.)