Return to Question

Updated on Nov.13.2024

D. New series for $\log(2)$

Finally, this one found (and proven) using WZ pairs

It has a binary splitting cost $C_s=1.21890$

It is interesting that these currently known fastest $\log(2)$ series, Eqs.(3), (5), (6) and (7), all are proven by means of Wilf-Zeilberger $(F,G)$ pairs starting from the same WZ seed and base companion $F(n,k)$ by using a method indicated in this MO question. The proofs are placed as an update to my answer below.

Updated on Nov.13 and Nov.21.2024

D. New series for $\log(2)$ and $\log(3)$

Also, this one was found (and proven) using WZ pairs

It has a binary splitting cost $C_s=1.21890$

Finally, for $\log(3)$, the following series was found with PSLQ. It has a cost $C_s=1.64589...$. This series performs slightly better than Machin-like arcotanh formulas being the 2nd fastest series currently known. $$\begin{equation*}\small{\log(3)}=\small{\sum_{n=1}^\infty\left(\frac{3}{2^4\cdot5^5}\right)^n\cdot\frac{P(n)}{n(2n-1)(6n-1)(6n-5)}\cdot\left[\begin{matrix} 1&\frac{1}{2}&\frac{1}{6}&\frac{5}{6}\\ \frac{1}{10}&\frac{3}{10}&\frac{7}{10}&\frac{9}{10}\\ \end{matrix}\right]_n}\tag{8}\label{8} \end{equation*}$$ where $$\small{P(n)}=\small{141168\,n^3 - 158016\,n^2 + 44804\,n - 3040}$$

It is interesting that these known fastest $\log(2)$ series, Eqs.(3), (5), (6) and (7), and $\log(3)$ series, Eqs. (1) and (8), are all proven by means of Wilf-Zeilberger $(F,G)$ pairs starting from the same WZ seed and base companion $F(n,k)$ by using a method indicated in this MO question. The proofs are placed as an update to my answer below.

Finally, this one

It is interesting that these currently known fastest $\log(2)$ series, Eqs.(3), (5), (6) and (7), all can be proven by means of Wilf-Zeilberger $(F,G)$ pairs starting from the same WZ seed and base companion $F(n,k)$ by using a method indicated in this MO question. The proofs are placed as an update to my answer below.

Finally, this one found (and proven) using WZ pairs

It is interesting that these currently known fastest $\log(2)$ series, Eqs.(3), (5), (6) and (7), all are proven by means of Wilf-Zeilberger $(F,G)$ pairs starting from the same WZ seed and base companion $F(n,k)$ by using a method indicated in this MO question. The proofs are placed as an update to my answer below.

$$\begin{equation*}\small{\log(2)}=\small{\sum_{n=1}^\infty\left(\frac{1}{2^4\cdot3^3\cdot7^7}\right)^n\cdot\frac{P(n)}{Q(n)}\cdot\left[\begin{matrix} 1&\frac12&\frac14&\frac34&\frac16&\frac56\\ \frac1{14}&\frac3{14}&\frac5{14}&\frac9{14}&\frac{11}{14}&\frac{13}{14}\\ \end{matrix}\right]_n}\tag{7}\label{7} \end{equation*}$$ where $$\small{P(n)}=\small{81969540480\,n^5 - 169950180480\,n^4 + 126495134424\,n^3 - 40884797604\,n^2 + 5510613042\,n - 226846575}$$$$\small{Q(n)}=\small{4n(2n-1)(4n-1)(4n-3)(6n-1)(6n-5)}$$

$$\begin{equation*}\small{\log(2)}=\small{\sum_{n=1}^\infty\left(\frac{1}{2^4\cdot3^3\cdot7^7}\right)^n\cdot\frac{P(n)}{Q(n)}\cdot\left[\begin{matrix} 1&\frac12&\frac14&\frac34&\frac16&\frac56\\ \frac1{14}&\frac3{14}&\frac5{14}&\frac9{14}&\frac{11}{14}&\frac{13}{14}\\ \end{matrix}\right]_n}\tag{7}\label{7} \end{equation*}$$ where $$\small{P(n)}=\small{81969540480\,n^5 - 169950180480\,n^4 + 126495134424\,n^3 }$$$$\small{- 40884797604\,n^2 + 5510613042\,n - 226846575}$$$$\small{Q(n)}=\small{4n(2n-1)(4n-1)(4n-3)(6n-1)(6n-5)}$$