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Infinite Geometric Series



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How do you know if a geometric series converges or diverges???

The Formula

The geometric series $$\sum\limits_{n=0}^{\infty}ar^{n}$$ converges when $$|r|\le 1$$ and sums to $$\frac{a}{1-r}$$

Examples

$$ \sum\limits_{n=0}^{\infty}3(\frac{3}{2})^{n} $$
Since $$ r \ge 1$$ , this series diverges .
$$ \sum\limits_{n=0}^{\infty}5(-1.05)^{n} $$
Since $$ r \ge 1$$ , this series also diverges .
$$ \sum\limits_{n=0}^{\infty}\frac{7}{3}(\frac{1}{3})^{n} $$
$$ \frac{\frac{7}{3}}{1-\frac{1}{3}} = \boxed{ \frac{7}{2} } $$ Since $$ r \lt 1 $$ , this series converges .
$$ \sum\limits_{n=1}^{\infty}\frac{1}{e^{n}} $$
Remember $$ \sum\limits_{n=1}^{\infty}\frac{1}{e^{n}} \text{can be rewritten} \sum\limits_{n=1}^{\infty}1(\frac{1}{e})^{n} $$
$$ r \le 1$$, converges, $$\frac{\frac{1}{e}}{1-\frac{1}{e}}= \boxed{ \frac{1}{e-1} } $$