Orion1

Higgs vacuum expectation value for the Standard Model: (ref. 1, pg. 4) [math]v_h = \sqrt{\frac{(\hbar c)^3}{\sqrt{2} G_F}}[/math] Higgs boson mass for the Standard Model: (ref. 1, pg. 4) [math]m_H = \sqrt{\frac{\lambda_h}{2}} v_h = \sqrt{\frac{\lambda_h (\hbar c)^3}{2 \sqrt{2} G_F}}[/math] Fermi coupling constant: (ref. 1, pg. 4),(ref. 2) [math]\frac{G_F}{(\hbar c)^3} = \frac{\sqrt{2}}{8} \frac{g_w^2}{m_W^2} = \frac{1}{\sqrt{2} v_h^2}[/math] Higgs vacuum expectation value: [math]\boxed{v_h = \frac{2 m_W}{g_w}}[/math] Integration via substitution: [math]m_H = \sqrt{\frac{\lambda_h}{2}} v_h = \sqrt{\frac{\lambda_h}{2}} \left( \frac{2 m_W}{g_w} \right) = \frac{\sqrt{2 \lambda_h} m_W}{g_w}[/math] Higgs boson mass: [math]\boxed{m_H = \frac{\sqrt{2 \lambda_h} m_W}{g_w}}[/math] Reference: Higgs Bosons: Theory And Searches - Fermi National Accelerator Laboratory Fermi coupling constant - Wikipedia Higgs boson - Wikipedia Higgs vacuum expectation value - Wikipedia W and Z bosons - Wikipedia Vector Boson Decay - ATLAS

Orion1 Higgs boson mass: [math]m_H = 123.111 \; \frac{\text{Gev}}{\text{c}^2}[/math] CERN Higgs boson mass: [math]m_H = 125.3 \pm 0.6 \; \frac{\text{Gev}}{\text{c}^2} \; \; \; \; \; \; 4.9 \; \sigma[/math] ATLAS Higgs boson mass: [math]m_H = 126.5 \; \frac{\text{Gev}}{\text{c}^2} \; \; \; \; \; \; 5 \; \sigma[/math] According to reference 3, pg. 4, the Higgs vacuum expectation value for the Standard Model is defined as: [math]v_h = \sqrt{\frac{(\hbar c)^3}{\sqrt{2} G_F}}[/math] According to reference 3, pg. 4, the mass of the Higgs boson for the Standard Model is defined as: [math]m_H = \sqrt{\frac{\lambda_h}{2}} v_h = \sqrt{\frac{\lambda_h (\hbar c)^3}{2 \sqrt{2} G_F}}[/math] [math]\boxed{m_H = \sqrt{\frac{\lambda_h (\hbar c)^3}{2 \sqrt{2} G_F}}}[/math] Higgs self-coupling parameter definition for lambda: [math]\lambda_h = 2 \left( \frac{m_H}{v_h} \right)^2[/math] Orion1 lambda: [math]\lambda_h = 2 \left( \frac{123.111}{246.221} \right)^2 = 0.5[/math] CERN lambda: [math]\lambda_h = 2 \left( \frac{125.3}{246.221} \right)^2 = 0.517943[/math] ATLAS lambda: [math]\lambda_h = 2 \left( \frac{126.5}{246.221} \right)^2 = 0.527911[/math] Reference: Higgs boson - Wikipedia Higgs vacuum expectation value - Wikipedia Higgs Bosons: Theory And Searches - Fermi National Accelerator Laboratory

Madhava-Leibniz series: (14th century) [Math]\pi = \sqrt{12} \sum_{k=0}^{\infty} \frac{(-3)^{-k}}{2k + 1}[/Math] Integer sum: [math]k \; (0,2,4,5,7,8,11,13,14)[/math] Reference: Madhava-Leibniz series - Wikipedia Chudnovsky algorithm: (the fastest algorithm in its class) [math]\frac{1}{\pi} = 12 \sum^\infty_{k=0} \frac{(-1)^k (6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 640320^{3k + 3/2}}[/math] Integer sum: [math]k \; (0,0,0,0,0,0,0,0,0)[/math] [math]n_{19} = 7[/math] at [math]k = 0[/math] [math]n_{28} = 3[/math] at [math]k = 1[/math] Mathematica source code: N[1/(12*Sum[((-1)^k*Factorial[6*k]*(13591409 + 545140134*k))/(Factorial[3*k]*Factorial[k]^3*640320^(3*k + 3/2)), {k, 0, 1}]), 30] Reference: Chudnovsky algorithm - Wikipedia

I agree, the fault is mine for selecting the Maclaurin series for arctan(x) for my initial evaluation, and that further discussion on this series needs to be suppressed because it represents a massive sidetrack. [math]\pi = \sum_{n=0}^{\infty} \frac{1}{16^n} \left( \frac{4}{8n+1} - \frac{2}{8n+4} - \frac{1}{8n+5} - \frac{1}{8n+6} \right)[/math] Integer sum: [math]n \; (0,0,1,1,2,3,3,4,5)[/math]

If the computer is evaluating pi from scratch, such as the case example for the Atn(x) and partial sum functions, then the computer does not know that the answer is pi, so how can it 'know' to round up to the nearest decimal point? and in the case where the solution is only 3.141 at that point in the sum, then there are no further decimal places to round up from!.

[math]4 \sum_{n=0}^{1687} \frac{(-1)^n}{2n+1} = 3.141... \; \; \; l = 4[/math] According to my computer algorithm and Mathematica and Wolframalpha, the solution is 3.141. Can anyone else verify? Reference: Wolframalpha - sum formula

[math]\pi = 4 \sum_{n=0}^{\infty} \frac{(-1)^n}{2n+1}[/math] I calculated the integer sum steps required to complete each numerical place, note that these numbers are fundamental to this equation and are independent of any computer specifications. (i.e. CPU, RAM, Memory, OS, basic language, etc.) Integer sum: [math]n \; ( \text{2}, \text{18}, \text{118}, \text{1687}, \text{10793}, \text{136120}, \text{1530011}, \text{18660287}, \text{155974698})[/math] Numerical place: [math]n_l \; (3,1,4,1,5,9,2,6,5)[/math] Numerical length: [math]l \; (1,2,3,4,5,6,7,8,9)[/math] Example: [math]4 \sum_{n=0}^{1687} \frac{(-1)^n}{2n+1} = 3.141... \; \; \; l = 4[/math] Note that [math]l = 9[/math] required a compiler.

[math]4 \cdot \sum_{n=0}^\infty \frac {(-1)^n} {2n+1} = \pi[/math] This equation appears to be the most reduced, however the arctan identity is gone.

This simple trigonometry equation trumps all the hyperbole calculus equations stated on this thread so far. Calculation for pi: [math]4 \cdot \arctan(1) = \pi[/math]

My calculation for the Higgs boson mass for the Standard Model. The Higgs boson mass is equal to one-half the Higgs vacuum expectation value. Higgs boson mass: [math]m_H = \frac{v_h}{2} = \frac{1}{2} \sqrt{\frac{(\hbar c)^3}{\sqrt{2} G_F}} = 123.111 \; \frac{\text{Gev}}{\text{c}^2}[/math] [math]\boxed{m_H = \frac{1}{2} \sqrt{\frac{(\hbar c)^3}{\sqrt{2} G_F}}}[/math] CERN Higgs boson mass: [math]m_H = 125.3 \pm 0.6 \; \frac{\text{Gev}}{\text{c}^2}[/math] Which implies that the Higgs boson achieves mass from the Higgs field vacuum via the Higgs mechanism. Reference: Physical constants - Wikipedia Higgs boson - Wikipedia Higgs mechanism - Wikipedia Higgs vacuum expectation value - Wikipedia Vector Boson Decay - ATLAS