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geometric series (Definition)

A geometric series is a series of the form

$\displaystyle \sum_{i=1}^n ar^{i-1}$    

(with $ a$ and $ r$ real or complex numbers). The partial sums of a geometric series are given by

$\displaystyle s_n=\sum_{i=1}^n ar^{i-1} = \frac{a(1 -r^n)}{1-r}.$ (1)

An infinite geometric series is a geometric series, as above, with $ n \rightarrow \infty$. It is denoted by

$\displaystyle \sum_{i=1}^\infty ar^{i-1}$    

If $ \vert r\vert\ge 1$, the infinite geometric series diverges. Otherwise it converges to

$\displaystyle \sum_{i=1}^\infty ar^{i-1} = \frac{a}{1-r}$ (2)

Taking the limit of $ s_n$ as $ n \rightarrow \infty$, we see that $ s_n$ diverges if $ \vert r\vert \ge 1$. However, if $ \vert r\vert < 1$, $ s_n$ approaches (2).

One way to prove (1) is to take

$\displaystyle s_n = a + ar + ar^2 + \cdots + ar^{n-1}$    

and multiply by $ r$, to get

$\displaystyle r s_n = ar + ar^2 + ar^3 + \cdots + ar^{n-1} + ar^{n}$    

subtracting the two removes most of the terms:

$\displaystyle s_n - rs_n = a - ar^n$    

factoring and dividing gives us

$\displaystyle s_n = \frac{a(1 -r^n)}{1-r}$    

$ \square$



"geometric series" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: geometric sequence, example of analytic continuation

Also defines:  infinite geometric series
Keywords:  infinite series
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Cross-references: terms, limit, converges, diverges, partial sums, complex numbers, real, series
There are 26 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 16087 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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