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Flashcards
\(e^x\)
\({1+\frac{x}{1!}+\frac{x²}{2!}+\dots+\frac{x^n}{n!}+\mathcal{O}(x^{n+1})}\)
\(\sinh(x)\)
\( {x+\frac{x^3}{3!}+\dots+\frac{x^{2n+1}}{(2n+1)!}+\mathcal{O}(x^{2n+3})}\)
\(\cosh(x)\)
\({1+\frac{x^2}{2!}+\frac{x^4}{4!}+\dots+\frac{x^{2n}}{(2n)!}+\mathcal{O}(x^{2n+2})}\)
\(\sin(x)\)
\( {x-\frac{x^3}{3!}+\frac{x^5}{5!}-\dots+(-1)^n\frac{x^{2n+1}}{(2n+1)!}+\mathcal{O}(x^{2n+3})}\)
\(\cos(x)\)
\({1-\frac{x^2}{2!}+\frac{x^4}{4!}+\dots+(-1)^n\frac{x^{2n}}{(2n)!}+\mathcal{O}(x^{2n+2})}\)
\((1+x)^\alpha\)
\(1+\alpha x+\frac{\alpha(\alpha -1)}{2!}x^2+\dots+\frac{\alpha(\alpha -1)\dots(\alpha -n+1)}{n!}+\mathcal{O}(x^{n+1})\)
\(\frac{1}{1-x}\)
\(1+x+x²+x³+\dots+x^n+\mathcal{O}(x^{n+1})\)
\(\ln(1-x)\)
\(-x-\frac{x²}{2}-\frac{x³}{3}-\dots-\frac{x^n}{n}+\mathcal{O}(x^{n+1})\)
\(\frac{1}{1+x}\)
\(1-x+x²-x³+\dots+(-1)^nx^n+\mathcal{O}(x^{n+1})\)
\(\ln(1+x) \)
\(x-\frac{x²}{2}+\frac{x³}{3}-\dots+(-1)^{n-1}\frac{x^n}{n}+\mathcal{O}(x^{n+1})\)
\(\sqrt{1+x}\)
\(1+\frac{x}{2}-\frac{x²}{8}+\dots+(-1)^{n-1}\frac{1\times3\times\dots\times(2n-3)}{2\times4\times\dots\times2n}x^n+\mathcal{O}(x^{n+1})\)
\(\frac{1}{\sqrt{1+x}}\)
\(1-\frac{1}{2}x-\frac{3}{8}x²+\dots+(-1)^{n-1}\frac{1\times3\times\dots\times(2n-1)}{2\times4\times\dots\times2n}x^n+\mathcal{O}(x^{n+1})\)
\(\tan(x)\)
\(x+\frac{1}{3}x³+\frac{2}{15}x^5+\frac{17}{315}x^7+\mathcal{O}(x⁹)\)
