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The Taylor polynomials for ln(1 + x) only provide accurate approximations in the range −1 < x ≤ 1. For x > 1, Taylor polynomials of higher degree provide worse approximations. The Taylor approximations for ln(1 + x) (black). For x > 1, the approximations diverge. Pictured is an accurate approximation of sin x around the point x = 0. The ...
The figure is a graph of ln(1 + x) and some of its Taylor polynomials around 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside this region, the higher-degree Taylor polynomials devolve to worse approximations for the function.
The approximation can be proven several ways, and is closely related to the binomial theorem. By Bernoulli's inequality , the left-hand side of the approximation is greater than or equal to the right-hand side whenever x > − 1 {\displaystyle x>-1} and α ≥ 1 {\displaystyle \alpha \geq 1} .
The table compares exact values of π(x) to the two approximations x / log x and li(x). The approximation difference columns are rounded to the nearest integer, but the "% error" columns are computed based on the unrounded approximations. The last column, x / π(x), is the average prime gap below x.
For example, with z = 1.5 the third approximation yields 0.4167, which is about 0.011 greater than ln(1.5) = 0.405465, and the ninth approximation yields 0.40553, which is only about 0.0001 greater. The n th partial sum can approximate ln( z ) with arbitrary precision, provided the number of summands n is large enough.
The simple formula for the factorial, x! = 1 × 2 × ⋯ × x is only valid when x is a positive integer, and no elementary function has this property, but a good solution is the gamma function () = (+). [1]
ln(r) is the standard natural logarithm of the real number r. Arg(z) is the principal value of the arg function; its value is restricted to (−π, π]. It can be computed using Arg(x + iy) = atan2(y, x). Log(z) is the principal value of the complex logarithm function and has imaginary part in the range (−π, π].
An approximation of the average number of divisors of all numbers from 1 to a given n. [32] The Lenstra–Pomerance–Wagstaff conjecture on the frequency of Mersenne primes. [33] An estimation of the efficiency of the euclidean algorithm. [34] Sums involving the Möbius and von Mangolt function.