Intended for advanced undergraduates and graduate students, this concise text focuses on the convergence of real series. Definitions of the terms and summaries of those results in analysis that are of special importance in the theory of series are specified at the outset. In the interests of maintaining a succinct presentation, discussion of the question of the upper and lower limits of a function is confined to an outline of those properties with a direct bearing on the convergence of series.
The central subject of this text is the convergence of real series, but series with complex terms and real infinite products are also examined as illustrations of the main theme. Infinite integrals appear only in connection with the integral test for convergence. Topics include functions and limits, real sequences and series, series of non-negative terms, general series, series of functions, the multiplication of series, infinite products, and double series. Prerequisites include a familiarity with the principles of elementary analysis.
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absolutely convergent aﬁ analysis binomial series bn+1 bounded function convergence the series convergent or properly convergent series converges uniformly deﬁned deﬁnition denote divergent series diverges to zero double series example ﬁnd ﬁnite limit ﬁnite number ﬁnite sum ﬁrst ﬁxed following theorem follows at once follows from Theorem Gamma functions geometry given Hence inequality inﬁnite product inﬁnite series integer interval introduction lim f(x mathematical monotonic decreasing function MULTIPLICATION OF SERIES negative integer non-negative terms obtain oscillate ﬁnitely positive integral values positive number power series problems product H properly divergent Prove quantum mechanics radius of convergence Ratio test real number real series repeated series result follows satisﬁed sequence series is convergent series of non-negative series whose terms series Z'a sinh suﬁicient tend to zero tends to inﬁnity term by term text covers Theorem 18 Theorem 39 theory uniformly convergent values of ac vergent write