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geometric series
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(Definition)
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A geometric series is a series of the form
(with $a$ and $r$ real or complex numbers). The partial sums of a geometric series are given by
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(1) |
An infinite geometric series is a geometric series, as above, with $n \rightarrow \infty$ . It is denoted by
If $|r|\ge 1$ , the infinite geometric series diverges. Otherwise it converges to
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(2) |
Taking the limit of $s_n$ as $n \rightarrow \infty$ , we see that $s_n$ diverges if $|r| \ge 1$ . However, if $|r| < 1$ , $s_n$ approaches (2).
One way to prove (1) is to take
and multiply by $r$ , to get
subtracting the two removes most of the terms:
factoring and dividing gives us
$\square$
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"geometric series" is owned by mathcam. [ full author list (2) | owner history (1) ]
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Cross-references: terms, limit, converges, diverges, partial sums, complex numbers, real, series
There are 34 references to this entry.
This is version 12 of geometric series, born on 2002-01-03, modified 2006-10-25.
Object id is 1188, canonical name is GeometricSeries.
Accessed 20384 times total.
Classification:
| AMS MSC: | 40A05 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences) |
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Pending Errata and Addenda
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