| Version current |
Version 11 |
| A \emph{geometric series} is a series of the form |
A \emph{geometric series} is a series of the form |
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| \begin{align*} |
\begin{align*} |
| \sum_{i=1}^n ar^{i-1} |
\sum_{i=1}^n ar^{i-1} |
| \end{align*} |
\end{align*} |
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| (with $a$ and $r$ real or complex numbers). The partial sums of a geometric series are given by |
(with $a$ and $r$ real or complex numbers). The partial sums of a geometric series are given by |
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| \begin{align} |
\begin{align} |
| s_n=\sum_{i=1}^n ar^{i-1} = \frac{a(1 -r^n)}{1-r}. |
s_n=\sum_{i=1}^n ar^{i-1} = \frac{a(1 -r^n)}{1-r}. |
| \end{align} |
\end{align} |
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| An \emph{infinite geometric series} is a geometric series, as above, with $n \rightarrow \infty$. It is denoted by |
An \emph{infinite geometric series} is a geometric series, as above, with $n \rightarrow \infty$. It is denoted by |
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| \begin{align*} |
\begin{align*} |
| \sum_{i=1}^\infty ar^{i-1} |
\sum_{i=1}^\infty ar^{i-1} |
| \end{align*} |
\end{align*} |
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| If $|r|\ge 1$, the infinite geometric series diverges. Otherwise it converges to |
If $|r|\ge 1$, the infinite geometric series diverges. Otherwise it converges to |
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| \begin{align} |
\begin{align} |
| \sum_{i=1}^\infty ar^{i-1} = \frac{a}{1-r} |
\sum_{i=1}^\infty ar^{i-1} = \frac{a}{1-r} |
| \end{align} |
\end{align} |
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| 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). |
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). |
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| One way to prove (1) is to take |
One way to prove (1) is to take |
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| \begin{align*} |
\begin{align*} |
| s_n = a + ar + ar^2 + \cdots + ar^{n-1} |
s_n = a + ar + ar^2 + \cdots + ar^{n-1} |
| \end{align*} |
\end{align*} |
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| and multiply by $r$, to get |
and multiply by $r$, to get |
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| \begin{align*} |
\begin{align*} |
|
r s_n = ar + ar^2 + ar^3 + \cdots + ar^{n-1} + ar^{n}
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r s_n = ar + ar^2 + ar^3 + \cdots + +ar^{n-1} + ar^{n}
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| \end{align*} |
\end{align*} |
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| subtracting the two removes most of the terms: |
subtracting the two removes most of the terms: |
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| \begin{align*} |
\begin{align*} |
| s_n - rs_n = a - ar^n |
s_n - rs_n = a - ar^n |
| \end{align*} |
\end{align*} |
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| factoring and dividing gives us |
factoring and dividing gives us |
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| \begin{align*} |
\begin{align*} |
| s_n = \frac{a(1 -r^n)}{1-r} |
s_n = \frac{a(1 -r^n)}{1-r} |
| \end{align*} |
\end{align*} |
| $\square$ |
$\square$ |
