Last updates: 7 December 2009

Write out the first four terms of the series
$\sum _{n=0}^{\infty}\frac{{x}^{n}}{n+1}$. | |

Write out the first four terms of the series
$\sum _{n=0}^{\infty}\frac{{x}^{n}}{n\text{!}}$. | |

Write out the first four terms of the series
$\sum _{n=1}^{\infty}\frac{{(x-1)}^{n}}{n}$. | |

Write out the first four terms of the series
$\sum _{n=0}^{\infty}{\left(-1\right)}^{n}\frac{{x}^{n}}{n+2}$. | |

Find the Taylor expansion of
${e}^{x}$ at
$x=0$. | |

Find the Taylor expansion of
$\mathrm{sinh}x$ at
$x=0$. | |

Find the Taylor expansion of
$\frac{1}{1-x}$ at
$x=0$. | |

Find the Taylor expansion of
${e}^{x}$ at
$x=2$. | |

Find the Taylor expansion of
$\mathrm{log}x$ at
$x=1$. | |

Find the Taylor expansion of
$\frac{1}{{x}^{2}}$ at
$x=1$. | |

Prove the identity
${e}^{ix}=\mathrm{cos}x+i\mathrm{sin}x$. | |

Prove the identity
${e}^{x}=\mathrm{cosh}x+\mathrm{sinh}x$. | |

Find the radius of convergence of the series
$\sum _{n=0}^{\infty}\frac{{x}^{n}}{n+1}$. | |

Find the radius of convergence of the series
$\sum _{n=0}^{\infty}{\left(-1\right)}^{n}\frac{{(x+1)}^{n}}{{(n+1)}^{2}}$. | |

Find the radius of convergence of the series
$\sum _{n=0}^{\infty}\frac{{x}^{n}}{n\text{!}}$. | |

Find the radius of convergence of the series
$\sum _{n=1}^{\infty}\frac{{(2x-1)}^{n}}{\sqrt[3]{n}}$. | |

Find the interval of convergence of the series
$\sum _{n=0}^{\infty}\frac{{x}^{n}}{n+1}$. | |

Find the interval of convergence of the series
$\sum _{n=0}^{\infty}{\left(-1\right)}^{n}\frac{{(x+1)}^{n}}{{(n+1)}^{2}}$. | |

Find the interval of convergence of the series
$\sum _{n=0}^{\infty}\frac{{x}^{n}}{n\text{!}}$. | |

Find the interval of convergence of the series
$\sum _{n=1}^{\infty}\frac{{(2x-1)}^{n}}{\sqrt[3]{n}}$. | |

Find the sum of the series
$\sum _{n=1}^{\infty}n{x}^{n-1}$. | |

Find the sum of the series
$\sum _{n=0}^{\infty}\frac{{x}^{n+1}}{n+1}$. | |

Find the sum of the series
$\sum _{n=1}^{\infty}\frac{{x}^{n}}{n}$. | |

Find the sum of the series
$\sum _{n=1}^{\infty}\frac{n}{{3}^{n-1}}$. | |

Find the sum of the series
$\sum _{n=1}^{\infty}\frac{1}{n{2}^{n+1}}$. | |

Find the sum of the series
$\sum _{n=1}^{\infty}n(n-1){\left(\frac{1}{4}\right)}^{n}$. | |

Find the power series representation of
$\frac{1}{1+2x}$ and determine its radius of convergence. | |

Find the power series representation of
$\frac{1}{1+{x}^{2}}$ and determine its radius of convergence. | |

Find the power series representation of
$\frac{x}{1+x}$ and determine its radius of convergence. | |

Find the power series representation of
$\frac{1}{{(1+x)}^{2}}$ and determine its radius of convergence. | |

Find the power series representation of
$\mathrm{arctan}x$ and determine its radius of convergence. | |

Find the power series representation of
$\mathrm{log}(2+x)$ and determine its radius of convergence. | |

Find the power series representation of
$\int {e}^{{x}^{3}}dx$.
| |

Find the power series representation of
$\int \frac{\mathrm{sinh}x}{x}dx$. | |

Find an infinite series representation of
${\int}_{-1}^{1}\frac{\mathrm{sinh}x}{x}dx$. | |

Find an infinite series representation of
${\int}_{0}^{1}{e}^{{x}^{3}}dx$. |

[BG]
A. Braverman and
D. Gaitsgory,
* Crystals via the affine Grassmanian*,
Duke Math. J.
**107** no. 3, (2001), 561-575;
arXiv:math/9909077v2,
MR1828302 (2002e:20083)