More Articles from Archive ~ Analisi

$\displaystyle e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots,$ $\displaystyle a^x=1+x\log\, a+\frac{(x\log\, a)^2}{2!}+\frac{(x\log\, a)^3}{3!}+\cdots,$ $\displaystyle \log(1+x)=x-\frac{x^2}2+\frac{x^3}3-\frac{x^4}4+\cdots$, $\displaystyle \log(1-x)=x+\frac{x^2}2+\frac{x^3}3+\frac{x^4}4+\cdots$, $\displaystyle \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$, $\displaystyle \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots$, $\displaystyle \arctan x=x-\frac{x^3}3+\frac{x^5}5-\frac{x^7}7+\frac{x^9}9-\cdots$ $\displaystyle \arcsin x=x+\frac 12 \frac{x^3}3+\frac 12\frac 34\frac{x^5}5+\frac 12\frac 34\frac 56 \frac{x^7}7+\cdots$ $\displaystyle (1-x)^{-\frac 12}=1+\frac 12 x+ \frac 12 \frac 34 x^2+\frac 12\frac 34\frac56 x^3+\frac 12\frac 34\frac 56 \frac 78 x^4+\cdots$ $\displaystyle (1-x)^{-n}=1+n x+ \frac{n(n+1)}{2!} x^2+\frac […]

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