## Some generalizations in the theory of summable divergent series |

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Abel's lemma approaches its limit approaches its Pringsheim approaches its repeated Borel's function call it summable conditions 1°-5 conditions of consistency consistency is satisfied continuous functions convergent with sum defined definition of uniform double limit lim equal to sum equation exists finite number finite Sum follows at once given series term Hardy and Chapman Hence Infinite Series Let fv Let the function lim G lim lim h lim sn limit lim ap limit n lim limit uniformly Math method of summation monotonic monotonic function Nielsen non-convergent series parameter path F positive integral values positive values Pringsheim double limit properly divergent series Quarterly Journ repeated double limit Riesz Riesz function satisfies the conditions séries divergentes series is summable series is uniformly Stirling's formula Sum equal summation of non-convergent term by term theorem follows Theory of Infinite u)fv uniformly convergent series uniformly with respect upper bound variables