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References
K. Knopp, Über das Eulersche Summierungsverfahren I,Math. Zeitschr.,15 (1922), pp. 226–253; II,18 (1923), pp. 125–156.
W. Meyer-König, Die Umkehrung des Euler-Knoppschen und des Borelschen Limitierungsverfahrens auf Grund einer Lückenbedingung,Math. Zeitschr.,49 (1943–44), pp. 151–160.
P. Erdős,Acad. Serbe Sci. Publ. Inst. Math.,4 (1952), pp. 51–56. RecentlyMeyer-König proved his conjecture:W. Meyer-König, Bemerkung zu einem Lückenumkehrsatz von H. R. Pitt,Math. Zeitschr.,57 (1952–53), pp. 351–352.
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A similar lemma is used in a paper byMacintyre and myself,Edinburgh Math. Proc., Ser. 2,10 (1954).
This can be proved as follows: We have for\( > \frac{{x^2 }}{{2\log 2}}\left( {1 + \frac{x}{n}} \right)^n > \frac{1}{2}e^x (namely,{\text{ }}n{\text{ }}log\left( {1 + \frac{x}{n}} \right) > x - \frac{{x^2 }}{{2n}} > x - log2)\). Now we have by (2) and (3)l−j=o(n l−nj). Puttingx=K(l−j) 1/2, n=nl−nj, the result follows.
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To the memory ofO. Szász
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Erdős, P. On a high-indices theorem in Borel summability. Acta Mathematica Academiae Scientiarum Hungaricae 7, 265–281 (1956). https://doi.org/10.1007/BF02020523
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DOI: https://doi.org/10.1007/BF02020523