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This article examines a class of series convergence tests, known as the mth ratio tests, that were introduced by Sayel A. Ali in 2008. We pay particular attention to the case where m = 2. We also c...

(2008). Summing a Curious, Slowly Convergent Series. The American Mathematical Monthly: Vol. 115, No. 6, pp. 525-540.

The statement of the inequality is actually a little more precise, but we shall discuss that later. Carleman used the inequality as part of a new proof of a delightful theorem of Denjoy about quasi-analytic functions. We shall not pursue that application here since we are interested in the inequality as a theorem about infinite series. We wish to expound in detail some of the many proofs of this theorem. We remark in passing, with approval, that editors of mathematical journals in the 1920s were happy to publish many different proofs of the same theorem. There are many known results about the convergence (or divergence) of series obtained by linear transformations of a nonnegative sequence {an }. For example, if Ain is the arithmetic mean of a1, a2, ... , an, then Y An, always diverges unless {an } is the zero sequence (for the sum dominates part of the harmonic series). The arithmeticgeometric mean inequality (AGM) asserts that y, < An, with equality if and only if the sequence {an} is constant. So the AGM-inequality must be strict when E an is convergent and nonzero. It is perhaps surprising a priori that the AGM-inequality is so

and declare the series to converge provided that limN-,N SN exists as a finite real number, which number is called the sum of the given series. This paradigm becomes so ingrained in most of us that we seldom question its appropriateness. To be sure, a frequently useful theorem of K. Weierstrass (1815-1897) tells us that an absolutely convergent series may be rearranged at will, with the convergence undisturbed and the resulting sum the same. A companion theorem due to G. F. B. Riemann (1826-1866), asserting that a conditionally convergent series may be rearranged to obtain any sum in the extended reals, seems to lay to rest any desire to rearrange conditionally convergent series. But that desire has blossomed forth in the authors of this article. One of us learned, in the course of teaching an analysis class, that there exist permutations a of the positive integers N with the following two properties

One of the surprising results in an elementary calculus course is that a rearrangement of a conditionaly convergent series may change its sum, even its very convergence. Observing typical textbook examples of this phenomenon, it turns out that during the rearrangement some of the terms are moved arbitrarily large distances from their original locations. Is this necessary? The answer is positive. Indeed, we can assert

(2001). The Sum of Some Convergent Series. The American Mathematical Monthly: Vol. 108, No. 9, pp. 851-855.

(1977). Partial Sums of Infinite Series, and How They Grow. The American Mathematical Monthly: Vol. 84, No. 4, pp. 237-258.

(1914). A Formula for the Sum of a Certain Type of Infinite Power Series. The American Mathematical Monthly: Vol. 21, No. 9, pp. 292-296.

(1945). Convergent Monotone Series. The American Mathematical Monthly: Vol. 52, No. 2, pp. 70-72.

(1914). A Curious Convergent Series. The American Mathematical Monthly: Vol. 21, No. 2, pp. 48-50.