Fun With Fourier Series
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2008
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| _version_ | 1866908995892019200 |
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| author | Baillie, Robert |
| author_facet | Baillie, Robert |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_0806_0150 |
| institution | arXiv |
| publishDate | 2008 |
| record_format | arxiv |
| spellingShingle | Fun With Fourier Series Baillie, Robert Classical Analysis and ODEs 40-01, 42-01 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. |
| title | Fun With Fourier Series |
| topic | Classical Analysis and ODEs 40-01, 42-01 |
| url | https://arxiv.org/abs/0806.0150 |