Збережено в:
| Автор: | |
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| Формат: | Preprint |
| Опубліковано: |
2008
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| Предмети: | |
| Онлайн доступ: | https://arxiv.org/abs/0806.0150 |
| Теги: |
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Зміст:
- By using computers to do experimental manipulations on Fourier series, we construct additional series with interesting properties. We construct several series whose sums remain unchanged when the $n^{th}$ term is multiplied by $\sin(n)/n$. One example is this classic series for $π/4$: \[ \fracπ{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \dots = 1 \cdot \frac{\sin(1)}{1} - \frac{1}{3} \cdot \frac{\sin(3)}{3} + \frac{1}{5} \cdot \frac{\sin(5)}{5} - \frac{1}{7} \cdot \frac{\sin(7)}{7} + \dots . \] Another example is \[ \sum_{n=1}^{\infty} \frac{\sin(n)}{n} = \sum_{n=1}^{\infty} \left(\frac{\sin(n)}{n}\right)^2 = \frac{π-1}{2}. \] This paper also discusses an included Mathematica package that makes it easy to calculate and graph the Fourier series of many types of functions.