Timeline for Cosine of a Partial Sum
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Nov 1, 2023 at 2:15 | history | edited | Michael Hardy | CC BY-SA 4.0 |
added 180 characters in body
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| May 18, 2011 at 4:17 | vote | accept | Jackson Walters | ||
| Mar 27, 2013 at 18:40 | |||||
| May 16, 2011 at 8:46 | comment | added | Jackson Walters | Yup, there are a couple typos above, but I don't see a way to edit that comment. I don't have a strict definition for "closed-form expression", and apparently that is a pretty vague term in general. This is as precise as I can make it: an expression with finitely many symbols denoting elementary operations such as addition, multiplication, etc. So no, I do not consider $\displaystyle\sum_{n=0}^\infty$ a closed-form expression as it involves a limiting process. As for the above expression involving permutation of subsets of natural numbers, I would reject it as not being explicit enough. | |
| May 16, 2011 at 7:34 | answer | added | Jackson Walters | timeline score: 0 | |
| May 16, 2011 at 5:27 | comment | added | S. Carnahan♦ | The formula you gave in the comment is slightly wrong, since you need to change both appearances of $\theta$ to $a$, and you need to allow $A$ to contain the number $0$. Also, please specify a definition of "closed-form expression" that doesn't make the problem tautological or impossible. For example, do you consider the expression $\sum_n a_n$ closed-form? | |
| May 16, 2011 at 3:43 | answer | added | Michael Hardy | timeline score: 1 | |
| May 16, 2011 at 2:27 | comment | added | Jackson Walters | Well the only formula I've seen for $cos(\displaystyle\sum_{n=0}^\infty a_{n})$ is $\sum_{\text{even}\ k \ge 0} ~ (-1)^{k/2} ~~ \sum_{\begin{smallmatrix} A \subseteq \{\,1,2,3,\dots\,\} \\ \left|A\right| = k\end{smallmatrix}} \left(\prod_{i \in A} \sin\theta_i \prod_{i \not \in A} \cos\theta_i\right) $ which isn't really enumerative, and I don't think there should be anything standing in the way of an enumerative expression in the case with finitely many terms. | |
| May 16, 2011 at 1:23 | comment | added | Zen Harper | If you have an expression for cos of the sum to infinity, why not just let $a_j=0$ for all $j>m$? | |
| May 16, 2011 at 0:49 | history | asked | Jackson Walters | CC BY-SA 3.0 |