![The Number "E" [Natural Log] To Approximately 1 Million Places](/image/coverpic3d.php?art=book&size=xl&p=ao5d837745a1735_1404308692.0591299534)
The number e is an important mathematical constant that is the base of the natural logarithm. It is approximately equal to 2.71828,[1] and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series[2]
e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots
The constant can be defined in many ways; for example, e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is equal to 1.[3] The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case, e is the number whose natural logarithm is 1. There are also more alternative characterizations.
e = \displaystyle\sum\limits_{n = 0}^{ \infty} \dfrac{1}{n!} = 1 + \frac{1}{1} + \frac{1}{1\cdot 2} + \frac{1}{1\cdot 2\cdot 3} + \cdots
The constant can be defined in many ways; for example, e is the unique real number such that the value of the derivative (slope of the tangent line) of the function f(x) = ex at the point x = 0 is equal to 1.[3] The function ex so defined is called the exponential function, and its inverse is the natural logarithm, or logarithm to base e. The natural logarithm of a positive number k can also be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case, e is the number whose natural logarithm is 1. There are also more alternative characterizations.
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