Tiêu đề MATH Formula Handbook.pdf ✅

1. Series Arithmetic and Geometric progressions A.P. Sn = a + (a + d) + (a + 2d) + · · · + [a + (n − 1)d] = n 2 [2a + (n − 1)d] G.P. Sn = a + ar + ar 2 + · · · + ar n−1 = a 1 − r n 1 − r , S∞ = a 1 − r for |r| < 1 (These results also hold for complex series.) Convergence of series: the ratio test Sn = u1 + u2 + u3 + · · · + un converges as n → ∞ if limn→∞ un+1 un < 1 Convergence of series: the comparison test If each term in a series of positive terms is less than the corresponding term in a series known to be convergent, then the given series is also convergent. Binomial expansion (1 + x) n = 1 + nx + n(n − 1) 2! x 2 + n(n − 1)(n − 2) 3! x 3 + · · · If n is a positive integer the series terminates and is valid for all x: the term in x r is nCrx r or n r where nCr ≡ n! r!(n − r)! is the number of different ways in which an unordered sample of r objects can be selected from a set of n objects without replacement. When n is not a positive integer, the series does not terminate: the infinite series is convergent for |x| < 1. Taylor and Maclaurin Series If y(x) is well-behaved in the vicinity of x = a then it has a Taylor series, y(x) = y(a + u) = y(a) + u dy dx + u 2 2! d 2y dx 2 + u 3 3! d 3y dx 3 + · · · where u = x − a and the differential coefficients are evaluated at x = a. A Maclaurin series is a Taylor series with a = 0, y(x) = y(0) + x dy dx + x 2 2! d 2y dx 2 + x 3 3! d 3y dx 3 + · · · Power series with real variables e x = 1 + x + x 2 2! + · · · + x n n! + · · · valid for all x ln(1 + x) = x − x 2 2 + x 3 3 + · · · + (−1) n+1 x n n + · · · valid for −1 < x ≤ 1 cos x = e ix + e −ix 2 = 1 − x 2 2! + x 4 4! − x 6 6! + · · · valid for all values of x sin x = e ix − e −ix 2i = x − x 3 3! + x 5 5! + · · · valid for all values of x tan x = x + 1 3 x 3 + 2 15 x 5 + · · · valid for − π 2 < x < π 2 tan −1 x = x − x 3 3 + x 5 5 − · · · valid for −1 ≤ x ≤ 1 sin −1 x = x + 1 2 x 3 3 + 1.3 2.4 x 5 5 + · · · valid for −1 < x < 1 2