Timeline for In search of an alternative proof of a series expansion for $\log 2$
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 29, 2023 at 14:47 | comment | added | Wolfgang | It looks like for each $k$ there is a unique similar polynomial with rational coefficients for the denominator $\binom{kn}{k}\binom{2kn}{kn}2^{kn}$ (and alternating signs for odd $k$), e.g. for $k=4$ $$\log 2=\frac14\sum_{n=1}^{\infty} \frac{(14560 n^3 - 16176 n^2 + 5138 n - 417) }{\binom{4n}4\binom{8n}{4n}2^{4n}} $$ or for $k=5$: $$\log 2=\frac1{40}\sum_{n=1}^{\infty} \frac{(2275625n^4 - 3615750n^3 + 1935175n^2 - 395010n + 23544)(-1)^{n-1} }{\binom{5n}5\binom{10n}{5n}2^{5n}} $$ | |
| Aug 15, 2021 at 16:17 | vote | accept | T. Amdeberhan | ||
| Jul 18, 2021 at 19:51 | comment | added | Carlo Beenakker | these $\log 2$ formulas remind me of the binomial sums for $\pi$, such as $\pi = \sum_{n=0}^\infty \dfrac{50n-6}{{\displaystyle \tbinom{3n}{n}2^{n}}}$ | |
| Jul 18, 2021 at 16:13 | history | edited | T. Amdeberhan | CC BY-SA 4.0 |
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| Jul 18, 2021 at 15:06 | history | edited | T. Amdeberhan | CC BY-SA 4.0 |
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| Jul 18, 2021 at 14:49 | history | edited | T. Amdeberhan | CC BY-SA 4.0 |
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| Jul 18, 2021 at 8:26 | comment | added | Nemo | This is classical result due to Lehmer: Interesting Series Involving the Central Binomial Coefficient, D. H. Lehmer, The American Mathematical Monthly, Vol. 92, No. 7 (1985), pp. 449-457. | |
| Jul 17, 2021 at 20:08 | review | Close votes | |||
| Jul 20, 2021 at 13:00 | |||||
| Jul 17, 2021 at 19:54 | answer | added | Carlo Beenakker | timeline score: 15 | |
| Jul 17, 2021 at 19:50 | answer | added | Richard Stanley | timeline score: 10 | |
| Jul 17, 2021 at 19:46 | answer | added | Noam D. Elkies | timeline score: 13 | |
| Jul 17, 2021 at 19:37 | comment | added | T. Amdeberhan | Your reference gives formulas for series involving zeta functions. Can you point out anything about (2) in the reference? | |
| Jul 17, 2021 at 19:18 | history | asked | T. Amdeberhan | CC BY-SA 4.0 |