Extending tests for convergence of number series Elijah Liflyand, Sergey Tikhonov, and Maria Zeltser September, 2012 Liflyand et al Extending monotonicity September, 2012 1 / 28 In various well-known tests for convergence/divergence of number series 1 X ak; (1) k=1 with positive ak; monotonicity of the sequence of these ak is the basic assumption. Such series are frequently called monotone series. Tests by Abel, Cauchy, de la Vallee Poussin, Dedekind, Dirichlet, du Bois Reymond, Ermakov, Leibniz, Maclaurin, Olivier, Sapogov, Schl¨omilch are related to monotonicity. Goals and History Main Goal: Relax the monotonicity assumption for the sequence of terms of the series. Liflyand et al Extending monotonicity September, 2012 2 / 28 Such series are frequently called monotone series. Tests by Abel, Cauchy, de la Vallee Poussin, Dedekind, Dirichlet, du Bois Reymond, Ermakov, Leibniz, Maclaurin, Olivier, Sapogov, Schl¨omilch are related to monotonicity. Goals and History Main Goal: Relax the monotonicity assumption for the sequence of terms of the series. In various well-known tests for convergence/divergence of number series 1 X ak; (1) k=1 with positive ak; monotonicity of the sequence of these ak is the basic assumption. Liflyand et al Extending monotonicity September, 2012 2 / 28 Tests by Abel, Cauchy, de la Vallee Poussin, Dedekind, Dirichlet, du Bois Reymond, Ermakov, Leibniz, Maclaurin, Olivier, Sapogov, Schl¨omilch are related to monotonicity. Goals and History Main Goal: Relax the monotonicity assumption for the sequence of terms of the series. In various well-known tests for convergence/divergence of number series 1 X ak; (1) k=1 with positive ak; monotonicity of the sequence of these ak is the basic assumption. Such series are frequently called monotone series. Liflyand et al Extending monotonicity September, 2012 2 / 28 Goals and History Main Goal: Relax the monotonicity assumption for the sequence of terms of the series. In various well-known tests for convergence/divergence of number series 1 X ak; (1) k=1 with positive ak; monotonicity of the sequence of these ak is the basic assumption. Such series are frequently called monotone series. Tests by Abel, Cauchy, de la Vallee Poussin, Dedekind, Dirichlet, du Bois Reymond, Ermakov, Leibniz, Maclaurin, Olivier, Sapogov, Schl¨omilch are related to monotonicity. Liflyand et al Extending monotonicity September, 2012 2 / 28 T.J.I'a Bromwich, Introduction to the Theory of Infinite Series, MacMillan, New York, 1965. K. Knopp, Theory and Application of Infinite Series, Blackie& Son Ltd., London-Glasgow, 1928. G.M. Fichtengolz, Infinite Series: Rudiments, Gordon and Breach, New York, 1970. D.D. Bonar and M.J. Khoury, Real Infinite Series, MAA, Washington, DC, 2006. References Books: Liflyand et al Extending monotonicity September, 2012 3 / 28 K. Knopp, Theory and Application of Infinite Series, Blackie& Son Ltd., London-Glasgow, 1928. G.M. Fichtengolz, Infinite Series: Rudiments, Gordon and Breach, New York, 1970. D.D. Bonar and M.J. Khoury, Real Infinite Series, MAA, Washington, DC, 2006. References Books: T.J.I'a Bromwich, Introduction to the Theory of Infinite Series, MacMillan, New York, 1965. Liflyand et al Extending monotonicity September, 2012 3 / 28 G.M. Fichtengolz, Infinite Series: Rudiments, Gordon and Breach, New York, 1970. D.D. Bonar and M.J. Khoury, Real Infinite Series, MAA, Washington, DC, 2006. References Books: T.J.I'a Bromwich, Introduction to the Theory of Infinite Series, MacMillan, New York, 1965. K. Knopp, Theory and Application of Infinite Series, Blackie& Son Ltd., London-Glasgow, 1928. Liflyand et al Extending monotonicity September, 2012 3 / 28 D.D. Bonar and M.J. Khoury, Real Infinite Series, MAA, Washington, DC, 2006. References Books: T.J.I'a Bromwich, Introduction to the Theory of Infinite Series, MacMillan, New York, 1965. K. Knopp, Theory and Application of Infinite Series, Blackie& Son Ltd., London-Glasgow, 1928. G.M. Fichtengolz, Infinite Series: Rudiments, Gordon and Breach, New York, 1970. Liflyand et al Extending monotonicity September, 2012 3 / 28 References Books: T.J.I'a Bromwich, Introduction to the Theory of Infinite Series, MacMillan, New York, 1965. K. Knopp, Theory and Application of Infinite Series, Blackie& Son Ltd., London-Glasgow, 1928. G.M. Fichtengolz, Infinite Series: Rudiments, Gordon and Breach, New York, 1970. D.D. Bonar and M.J. Khoury, Real Infinite Series, MAA, Washington, DC, 2006. Liflyand et al Extending monotonicity September, 2012 3 / 28 Tests In its initial form the Maclaurin-Cauchy integral test reads as follows: Consider a non-negative monotone decreasing function f defined on [1; 1): Then the series 1 X f(k) (2) k=1 converges if and only if the integral Z 1 f(t) dt 1 is finite. In particular, if the integral diverges, then the series diverges. Liflyand et al Extending monotonicity September, 2012 4 / 28 In its simplest form, it is given as follows. Let f be a continuous (this is not necessary) non-negative monotone decreasing function for t > 1: If for t large enough f(et)et=f(t) ≤ q < 1; then series (2) converges, while if f(et)et=f(t) ≥ 1; then series (2) diverges. In fact, it is known that one can get a family of tests by replacing et with a positive increasing function '(t) satisfying certain properties. We will extend such a more general assertion. Tests Ermakov's test. Liflyand et al Extending monotonicity September, 2012 5 / 28 Let f be a continuous (this is not necessary) non-negative monotone decreasing function for t > 1: If for t large enough f(et)et=f(t) ≤ q < 1; then series (2) converges, while if f(et)et=f(t) ≥ 1; then series (2) diverges. In fact, it is known that one can get a family of tests by replacing et with a positive increasing function '(t) satisfying certain properties. We will extend such a more general assertion. Tests Ermakov's test. In its simplest form, it is given as follows. Liflyand et al Extending monotonicity September, 2012 5 / 28 If for t large enough f(et)et=f(t) ≤ q < 1; then series (2) converges, while if f(et)et=f(t) ≥ 1; then series (2) diverges. In fact, it is known that one can get a family of tests by replacing et with a positive increasing function '(t) satisfying certain properties. We will extend such a more general assertion. Tests Ermakov's test. In its simplest form, it is given as follows. Let f be a continuous (this is not necessary) non-negative monotone decreasing function for t > 1: Liflyand et al Extending monotonicity September, 2012 5 / 28 while if f(et)et=f(t) ≥ 1; then series (2) diverges. In fact, it is known that one can get a family of tests by replacing et with a positive increasing function '(t) satisfying certain properties. We will extend such a more general assertion. Tests Ermakov's test. In its simplest form, it is given as follows. Let f be a continuous (this is not necessary) non-negative monotone decreasing function for t > 1: If for t large enough f(et)et=f(t) ≤ q < 1; then series (2) converges, Liflyand et al Extending monotonicity September, 2012 5 / 28 In fact, it is known that one can get a family of tests by replacing et with a positive increasing function '(t) satisfying certain properties. We will extend such a more general assertion. Tests Ermakov's test. In its simplest form, it is given as follows. Let f be a continuous (this is not necessary) non-negative monotone decreasing function for t > 1: If for t large enough f(et)et=f(t) ≤ q < 1; then series (2) converges, while if f(et)et=f(t) ≥ 1; then series (2) diverges. Liflyand et al Extending monotonicity September, 2012 5 / 28 We will extend such a more general assertion. Tests Ermakov's test. In its simplest form, it is given as follows. Let f be a continuous (this is not necessary) non-negative monotone decreasing function for t > 1: If for t large enough f(et)et=f(t) ≤ q < 1; then series (2) converges, while if f(et)et=f(t) ≥ 1; then series (2) diverges. In fact, it is known that one can get a family of tests by replacing et with a positive increasing function '(t) satisfying certain properties. Liflyand et al Extending monotonicity September, 2012 5 / 28 Tests Ermakov's test. In its simplest form, it is given as follows. Let f be a continuous (this is not necessary) non-negative monotone decreasing function for t > 1: If for t large enough f(et)et=f(t) ≤ q < 1; then series (2) converges, while if f(et)et=f(t) ≥ 1; then series (2) diverges. In fact, it is known that one can get a family of tests by replacing et with a positive increasing function '(t) satisfying certain properties. We will extend such a more general assertion. Liflyand et al Extending monotonicity September, 2012 5 / 28 If fakg is a positive monotone decreasing null sequence, then the series (1) and 1 X k 2 a2k k=1 converge or diverge simultaneously. This assertion is a partial case of the following classical result due to Schl¨omilch. Let (1) be a series whose terms are positive and non-increasing, Tests The well known Cauchy condensation test states that Liflyand et al Extending monotonicity September, 2012 6 / 28 This assertion is a partial case of the following classical result due to Schl¨omilch. Let (1) be a series whose terms are positive and non-increasing, Tests The well known Cauchy condensation test states that If fakg is a positive monotone decreasing null sequence, then the series (1) and 1 X k 2 a2k k=1 converge or diverge simultaneously.