geometric_sequences_and_sums
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| Both sides previous revisionPrevious revisionNext revision | Previous revision | ||
| geometric_sequences_and_sums [2021/10/22 14:24] – [with Infinite Series (n이 무한대일 때)] hkimscil | geometric_sequences_and_sums [2024/10/09 08:14] (current) – [with Infinite Series (n이 무한대일 때)] hkimscil | ||
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| Line 48: | Line 48: | ||
| X_{n} & = & ar^{(n-1)} \\ | X_{n} & = & ar^{(n-1)} \\ | ||
| & & \text{where | & & \text{where | ||
| - | & & \text{ | + | & & \text{ |
| r^{(n-1)} & = & 0 \\ | r^{(n-1)} & = & 0 \\ | ||
| \therefore \text{ | \therefore \text{ | ||
| Line 99: | Line 99: | ||
| \sum_{n=0}^{\infty}(ar^n) & = & a \cdot \frac {(1 - r^{n})}{1-r} \\ | \sum_{n=0}^{\infty}(ar^n) & = & a \cdot \frac {(1 - r^{n})}{1-r} \\ | ||
| & & \text{when } \\ | & & \text{when } \\ | ||
| - | & & n -> \infty, |r| < 1, r \ne 0 \\ | + | & & n \rightarrow |
| - | r^{n} & = & | + | & & r^{n} = 0 \\ |
| - | \sum_{n=0}^{\infty}(ar^n) & = & a \cdot \left(\frac{1}{1-r}\right) | + | \therefore \; \; \sum_{n=0}^{\infty}(ar^n) & = & a \cdot \left(\frac{1}{1-r}\right) |
| \end{eqnarray*} | \end{eqnarray*} | ||
geometric_sequences_and_sums.1634880247.txt.gz · Last modified: 2021/10/22 14:24 by hkimscil