| Series | Comment |
|---|---|
| \(\frac{1}{1-x}=\sum\limits_{k\,=\,0}^{\infty}x^{k}=1+x+x^{2}+x^{3}+\cdots\) | Converges for \(\vert x\vert <1\). |
| \(\tan ^{-1}x=\sum\limits_{k\,=\,0}^{\infty }(-1) ^{k}\frac{x^{2k+1}}{2k+1}=x-\frac{x^{3}}{3}+\frac{x^{5}}{5} -\,\cdots\) | Converges on the interval \([-1,\,1]\). |
| \(e^{x}=\sum\limits_{k\,=\,0}^{\infty }\frac{x^{k}}{k!}=1+\frac{x}{1!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\cdots\) | Converges for all real numbers \(x\). |
| \(\sin x=\sum\limits_{k\,=\,0}^{\infty }\frac{(-1)^{k}\,x^{2k+1}}{(2k+1)!}=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+\cdots\) | Converges for all real numbers \(x\). |
| \(\cos x=\sum\limits_{k\,=\,0}^{\infty }\frac{(-1)^{k}x^{2k}}{(2k)!}=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+\cdots\) | Converges for all real numbers \(x\). |
| \(\ln (1+x)=\sum\limits_{k\,=\,0}^{\infty }\frac{(-1)^{k}\,x^{k+1}}{k+1}\) | Converges on the interval \((-1,\,1].\) |
| \((1+x)^{m}=\sum\limits_{k\,=\,0}^{\infty }{m\choose k} \,x^{k}\) | For convergence, see the theorem on page 620. |