By Gazzola F., Radulescu V.
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71825. This approximation can be improved by adding more terms until the desired accuracy is reached. 71828 · · · . 3 The Exponential Function ex Raising e to x power, we have ex = lim n→∞ Let nx = m, then equation becomes x m = 1 n. 1+ 1 n nx . As n goes to ∞, so does m. Thus the above ex = lim m→∞ 1+ x m m . Now m may not be an integer, but the binomial formula is equally valid for noninteger power (one of the early discoveries of Isaac Newton). 13), we can express the exponential function as an inﬁnite series, x3 x4 x2 + + + ··· .
N n(n − 1)(n − 2) x 3 x n + +···+ 3! n n (1 − 1/n) 2 (1 − 1/n)(1 − 2/n) 3 x x + x +··· = 1+x+ 2! 3! n = 1+n Now as n → ∞, k n n . → 0. Therefore lim n→∞ 1+ x n n =1+x+ x3 x4 x2 + + + · · ·. 2! 3! 4! 13) becomes an inﬁnite series. Standard tests for convergence show that this is a convergent series for all real values of x. In other words, the value of (1 + nx )n does settle on a speciﬁc limit as n increase without bound. 2. 13) for the case of x = 1, e = lim n→∞ 1+ 1 n n =1+1+ 1 1 1 + + + ··· .
4 √ i3θ+ik2π 3 (1 − i) = 2 2e , k = 0, 1, 2, . . √ 4 3/2 i(3θ/2+kπ) (1 − i) = 8e , k = 0, 1. (1 − i) = (1 − i)3/2 = √ 4 3π , k=0 8 cos − 3π 8 + i sin − 8 √ 4 5π 5π 8 cos 8 + i sin 8 , k = 1. 2 Trigonometry and Complex Numbers Many trigonometric identities can be most elegantly proved with complex numbers. For example, taking the complex conjugate of the Euler formula (eiθ )∗ = (cos θ + i sin θ)∗ , we have e−iθ = cos θ − i sin θ. It is interesting to write this equation as e−iθ = ei(−θ) = cos(−θ) + i sin(−θ).