函數(shù)項(xiàng)級(jí)數(shù)一致收斂的判別-全文預(yù)覽
【正文】 明： 因, 故對(duì)任給的, (與無(wú)關(guān)), 使得當(dāng)時(shí), 對(duì)一切, 都有。因而。 余項(xiàng)判別法 定理 函數(shù)項(xiàng)級(jí)數(shù)在數(shù)集上一致收斂于的充要條件是。例2 試證在區(qū)間上一致收斂。例1 證明函數(shù)項(xiàng)級(jí)數(shù)，一致收斂。由于函數(shù)項(xiàng)級(jí)數(shù)的一致收斂性是由它的部分和數(shù)列來(lái)確定, 所以可以根據(jù)函數(shù)列一致收斂性定義得到等價(jià)定義。若級(jí)數(shù)(3)發(fā)散，則稱級(jí)數(shù)(1)在點(diǎn)發(fā)散。本文結(jié)合上述文獻(xiàn), 總結(jié)出了函數(shù)項(xiàng)級(jí)數(shù)一致收斂的其它判別法, 如對(duì)數(shù)判別法, 導(dǎo)數(shù)判別法, M判別法的推論等, 并給出了一些判別法的證明, 此外也用一些例題驗(yàn)證它的可行性。 文獻(xiàn)[1]討論了函數(shù)項(xiàng)級(jí)數(shù)一致收斂的基本判別法, 給出了一致收斂的定義和萊布尼茨判別法。函數(shù)項(xiàng)級(jí)數(shù)既可以被看作是對(duì)數(shù)項(xiàng)級(jí)數(shù)的推廣, 同時(shí)數(shù)項(xiàng)級(jí)數(shù)也可以看作是函數(shù)項(xiàng)級(jí)數(shù)的一個(gè)特例, 它們?cè)谘芯績(jī)?nèi)容上有許多相似之處。 uniform convergence。而因此本論文中提出了函數(shù)級(jí)數(shù)一致收斂的定義, 柯西一致收斂準(zhǔn)則, 魏爾斯特拉斯判別法(M判別法), 狄利克雷判別法, 阿貝爾判別法, 余項(xiàng)判別法, 積分判別法。 本文則在數(shù)項(xiàng)級(jí)數(shù)的基礎(chǔ)上, 分析函數(shù)項(xiàng)級(jí)數(shù)的收斂性定義及其判定, 函數(shù)項(xiàng)級(jí)數(shù)的分析性質(zhì)和函數(shù)的一致收斂有關(guān)。 Discrimination of uniform convergence of function seriesAbstract: The uniform convergence of function series is the concept of series of functions are the most basic and most important problem. In this paper, on the basis of a number of series, the definitions of convergence of function series and its decision, uniform convergence analysis of properties and functions related to the function of series. Therefore, this paper proposes a definition of uniform convergence of function series, Cauchy uniform convergence criteria the Weierstrass discrimination method (M identification method), Dirichlet discrimination law, Abel discriminant law, the remainder discriminant method, integration criterion method and article on the function series convergence discriminant method to promote mainly summarized Diagnostic Method derivative test, continuity discrimination law, forcing several discriminant method of convergence discrimination law and M inference of discrimination law, and apply function series consistent definition of convergence, it is important discrimination method and the necessary and sufficient conditions are given some proof of the conclusion of the paper.Keywords: Function Series。 而本文在給出這些判別法的同時(shí)并對(duì)函數(shù)項(xiàng)級(jí)數(shù)一致收斂的定義，柯西判別法，M判別法，阿貝爾判別法，萊布尼茲判別法加以補(bǔ)充和推廣，從而給判別函數(shù)項(xiàng)級(jí)數(shù)一致收斂提供了方便。 即函數(shù)項(xiàng)級(jí)數(shù)的一致收斂性。 文獻(xiàn)[10]對(duì)該問(wèn)題進(jìn)行了推廣, 得到了比試和根式判別法,
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