Formula Sheet \(f(x)\) Maclaurin Series Interval of Conve…

Fоrmulа Sheet (f(x)) Mаclаurin Series Interval оf Cоnvergence (dfrac{1}{1-x}) (displaystylesum_{n=0}^{infty}x^n) ((-1,1)) (ln(1-x)) (-displaystylesum_{n=1}^{infty}frac{x^n}{n}) ([-1,1)) (ln(1+x)) (displaystylesum_{n=1}^{infty}(-1)^{n+1}frac{x^n}{n}) ((-1,1]) (e^{x}) (displaystylesum_{n=0}^{infty}frac{x^n}{n!}) ((-infty,infty)) (cos x) (displaystylesum_{n=0}^{infty}(-1)^nfrac{x^{2n}}{(2n)!}) ((-infty,infty)) (sin x) (displaystylesum_{n=0}^{infty}(-1)^nfrac{x^{2n+1}}{(2n+1)!}) ((-infty,infty)) (arctan x) (displaystylesum_{n=0}^{infty}(-1)^nfrac{x^{2n+1}}{2n+1}) ([-1,1]) Taylor Remainder Estimate: If (left|f^{(N+1)}(c)right|le M) for all numbers (c) between (x) and (a), then the remainder (R_N(x)) of the Taylor series satisfies the inequality (displaystyle left|R_N(x)right|lefrac{Mleft|x-aright|^{N+1}}{(N+1)!}) for all numbers between (x) and (a). Taylor Coefficients: (displaystyle c_n=frac{f^{(n)}(a)}{n!})