What are some examples of infinite geometric series?

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Answer 1 · 2015-07-13T18:38:56

Here are some examples:

$1 + 1/2 + 1/4 + 1/8 + 1/16 +...$

$1 - 1 + 1 - 1 + 1 - 1 +...$

$1 + 2 + 4 + 8 + 16 +...$

All geometric series are of the form $sum_(i=0)^oo ar^i$ where $a$ is the initial term of the series and $r$ the ratio between consecutive terms.

In the three examples above, we have:

$a = 1$ , $r = 1/2$

$sum_(i=0)^oo ar^i = 2$

$a = 1$ , $r = -1$

$sum_(i=0)^oo ar^i$ does not converge - it alternates between $0$ and $1$ as each term is added.

$a = 1$ , $r = 2$

$sum_(i=0)^n ar^i -> oo$ as $n->oo$

The geometric series $sum_(i=0)^oo ar^i$ only converges in the following cases:

(1) $a = 0$

$sum_(i=0)^oo ar^i = 0$

(2) $abs(r) < 1$

$sum_(i=0)^oo ar^i = a/(1-r)$