## An Introduction to the Theory of Infinite SeriesThis edition consists largely of a reproduction of the first edition (which was based on lectures on Elementary Analysis given at Queen's College, Galway, from 1902-1907), with additional theorems and examples. Additional material includes a discussion of the solution of linear differential equations of the second order; a discussion of elliptic function formulae; expanded treatment of asymptomatic series; a discussion of trigonometrical series, including Stokes's transformation and Gibbs's phenomenon; and an expanded Appendix II that includes an account of Napier's invention of logarithms. |

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### Contents

CHAPTER I | 1 |

CHAPTER II | 26 |

CHAPTER III | 53 |

CHAPTER IV | 69 |

Double Series 7897 | 78 |

CHAPTER VI | 104 |

Examples | 114 |

CHAPTER VIII | 145 |

CHAPTER XI | 278 |

CHAPTER XII | 317 |

Asymptotic Series 324354 | 324 |

Trigonometrical Series 354387 | 354 |

APPENDIX I | 394 |

Examples | 430 |

APPENDIX II | 436 |

Some Theorems on Infinite Integrals and Gamma | 461 |

Special Power Series 170189 | 170 |

CHAPTER IX | 202 |

CHAPTER X | 227 |

Examples | 267 |

Examples | 514 |

MISCELLANEOUS EXAMPLES | 526 |

533 | |

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### Common terms and phrases

Abel's Lemma Abel's test Abel's theorem absolutely convergent apply asymptotic series binomial series calculation coefficients condition consider continuous function convergent series converges absolutely converges uniformly corresponding curve deduce definite limit denote differential equation Dirichlet's Dirichlet's test divergent series diverges double series easy equal Euler's example expansion expression finite follows formula fraction given gives greater Hence increases inequality integer integral interval less logarithmic Math method monotonic Napier negative notation number of terms obtain oscillates polynomial positive integer positive number positive terms power-series Pringsheim proof prove radius of convergence rational numbers result satisfied sequence series converges shew Similarly Sn(x steadily decreases suppose tends steadily tends to infinity tends to oo tends to zero term-by-term theorem Art theory uniform convergence upper limit variable verify Weierstrass's write