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Comparison of the convergence of the Wallis product (purple asterisks) and several historical infinite series for π. S n is the approximation after taking n terms. Each subsequent subplot magnifies the shaded area horizontally by 10 times.
The coincidence = =, correct to 2.4%, relates to the rational approximation , or / to within 0.3%. This relationship is used in engineering, for example to approximate a factor of two in power as 3 dB (actual is 3.0103 dB – see Half-power point ), or to relate a kibibyte to a kilobyte ; see binary prefix .
In that case the other envelope contains a/2 with probability 3/5 and 2a with probability 2/5. So either the first envelope contains 1, in which case the conditional expected amount in the other envelope is 2, or the first envelope contains a > 1, and though the second envelope is more likely to be smaller than larger, its conditionally ...
3.2 Efficient infinite series. 3.3 Other infinite series. 3.4 Machin-like formulae. 3.5 Infinite products. 3.6 Arctangent formulas. 3.7 Complex functions.
These definitions E 1, E 2, and E 3 of the envelope may be different sets. Consider for instance the curve y = x 3 parametrised by γ : R → R 2 where γ(t) = (t,t 3). The one-parameter family of curves will be given by the tangent lines to γ. First we calculate the discriminant . The generating function is
Bertrand's box paradox: the three equally probable outcomes after the first gold coin draw. The probability of drawing another gold coin from the same box is 0 in (a), and 1 in (b) and (c). Thus, the overall probability of drawing a gold coin in the second draw is 0 / 3 + 1 / 3 + 1 / 3 = 2 / 3 .
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