Home/Calculus II/Power series and Taylor series/Fundamental Maclaurin seriesSection 15.5Fundamental Maclaurin seriesFormulaCopy LaTeXex=∑n=0∞xnn!=1+x+x22!+x33!+⋯e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!} =1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdotsex=n=0∑∞n!xn=1+x+2!x2+3!x3+⋯View detail →FormulaCopy LaTeXsinx=∑n=0∞(−1)nx2n+1(2n+1)!=x−x33!+x55!−⋯\sin x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!} =x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdotssinx=n=0∑∞(−1)n(2n+1)!x2n+1=x−3!x3+5!x5−⋯View detail →FormulaCopy LaTeXcosx=∑n=0∞(−1)nx2n(2n)!=1−x22!+x44!−⋯\cos x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!} =1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdotscosx=n=0∑∞(−1)n(2n)!x2n=1−2!x2+4!x4−⋯View detail →FormulaCopy LaTeX11−x=∑n=0∞xn,∣x∣<1\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n, \qquad |x|<11−x1=n=0∑∞xn,∣x∣<1∣x∣<1|x|<1∣x∣<1View detail →FormulaCopy LaTeXln(1+x)=∑n=1∞(−1)n−1xnn,−1<x≤1\ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}, \qquad -1<x\le1ln(1+x)=n=1∑∞(−1)n−1nxn,−1<x≤1−1<x≤1-1<x\le1−1<x≤1View detail →FormulaCopy LaTeXarctanx=∑n=0∞(−1)nx2n+12n+1,∣x∣≤1\arctan x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1}, \qquad |x|\le1arctanx=n=0∑∞(−1)n2n+1x2n+1,∣x∣≤1∣x∣≤1|x|\le1∣x∣≤1View detail →with conditional convergence at x=±1x=\pm1x=±1 depending on the corresponding value.
FormulaCopy LaTeXex=∑n=0∞xnn!=1+x+x22!+x33!+⋯e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!} =1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdotsex=n=0∑∞n!xn=1+x+2!x2+3!x3+⋯View detail →
FormulaCopy LaTeXsinx=∑n=0∞(−1)nx2n+1(2n+1)!=x−x33!+x55!−⋯\sin x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!} =x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdotssinx=n=0∑∞(−1)n(2n+1)!x2n+1=x−3!x3+5!x5−⋯View detail →
FormulaCopy LaTeXcosx=∑n=0∞(−1)nx2n(2n)!=1−x22!+x44!−⋯\cos x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!} =1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdotscosx=n=0∑∞(−1)n(2n)!x2n=1−2!x2+4!x4−⋯View detail →
FormulaCopy LaTeX11−x=∑n=0∞xn,∣x∣<1\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n, \qquad |x|<11−x1=n=0∑∞xn,∣x∣<1∣x∣<1|x|<1∣x∣<1View detail →
FormulaCopy LaTeXln(1+x)=∑n=1∞(−1)n−1xnn,−1<x≤1\ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n}, \qquad -1<x\le1ln(1+x)=n=1∑∞(−1)n−1nxn,−1<x≤1−1<x≤1-1<x\le1−1<x≤1View detail →
FormulaCopy LaTeXarctanx=∑n=0∞(−1)nx2n+12n+1,∣x∣≤1\arctan x=\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{2n+1}, \qquad |x|\le1arctanx=n=0∑∞(−1)n2n+1x2n+1,∣x∣≤1∣x∣≤1|x|\le1∣x∣≤1View detail →