1 1 1/2! 1/3! ... 1/n！ The discovery that n tends to infinity equals e e e begins with differentiation. When h gradually approaches zero, the calculated value is infinitely close to a certain value 2.7 1828 ... This fixed value is E. The first person who discovered this value was Euler, a famous Swiss mathematician, who named this irrational number after the prefix E of his name. When calculating the derivative of logarithmic function, it is concluded that when a=e, the derivative of is 0. Therefore, it is reasonable to use logarithm based on e, which is called natural logarithm. If the exponential function ex is Taylor expansion, it is necessary to substitute x= 1 into the above formula to get the rapid convergence of this series. When the value of e approximates to 40 decimal places, it is necessary to extend the definition of exponential function ex to complex number z=x yi. Through the calculation of this series, we can get the de Morville theorem. The sum and difference angle formula of trigonometric function can be easily deduced. For example, Z 1 = X 1Y 1i, and Z 2 = X2Y2I. On the other hand, we can not only prove that e is irrational, but also a transcendental number, that is, it is not the root of any integer coefficient polynomial. This result was obtained by Hermite in 1873. Consider a discrete function (i.e. sequence) R, whose value u(n) at n is denoted as un. Usually we write this function as or (un). The difference of sequence u is still a sequence. The value it takes in n is defined as (for example): the difference sequence of sequence 1, 4, 8, 7, 6, -2, ... is 3, 4,-1, -8 ... Note: we say "series". However, it is very appropriate because it has a completely parallel analogy with continuous functions. Difference operator's properties (i) [collectively referred to as linearity] (ii) (constant) [fundamental theorem of difference equation] (iii) are among them, and (n) is called permutation sequence. (4) It is called natural geometric series. ㈣