Download common taylor series cheat sheet and more Cheat Sheet Mathematical Analysis in PDF only on Docsity! REPRESENTING FUNCTIONS USING POWER SERIES 1. The Taylor series of a function f(x) that is centered at x = a is the infinite series f(x) = ∞∑ n=0 fn(a)(x− a)n n! = f(a) + f ′(a)(x− a) 1! + f ′′(a)(x− a)2 2! + f ′′′(a)(x− a)3 3! + ... 2. The MacLaurin Series of a function is its Taylor series centered at a = 0. 3. The Maclaurin series for some basic functions. These are very important exam- ples, so you must memorize them. (a) 1 1−x = ∑ ∞ n=0 x n = 1 + x + x2 + x3 + ... (b) sin(x) = x 1! − x 3 3! + x 5 5! − x 7 7! + ... (c) cos(x) = 1− x 2 2! + x 4 4! − x 6 6! + ... (d) ex = 1 + x 1! + x 2 2! + x 3 3! + x 4 4! + ... Note: If you remember the series for sin(x), then the series for cos(x) can be obtained by simply taking its derivative. Furthermore, the power series expansion of a b+cx , sin(kx), cos(kx) and ekx can be obtained from expressions a,b,c, and d, by using the appropriate substitution. APPROXIMATING FUNCTIONS USING TAYLOR POLYNOMIALS Many modern calculators use Taylor polynomials, which are just Taylor series with only finitely many non-zero terms, to compute the values of ex, sin(x), etc. 1. the n-th degree Taylor polynomial of a function f is Tn(x) = f(a) + f ′(a)(x− a) 1! + f ′′(a)(x− a)2 2! + f ′′′(a)(x− a)3 3! + ... f (n)(a)(x− a)n n! 2. The accuracy of the Taylor polynomial Tn(x) over the interval (a− d, a+ d) can be computed via the ”Taylor Estimate” of the remainder function Rn(x): Rn(x) = |f(x)− Tn(x)| ≤ M |x− a|n+1 (n + 1)! Here M denotes the maximum value of f (n+1)(x) for |x− a| < d. 3. The power series of the function f converges to the value of the function, over the interval (a− d, a + d), if limn→∞Rn = 0. This is usually checked using the Taylor estimate(2) and the fact that lim n→∞ xn n! = 0.
