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<p>There is a natural connection between the finite sums of the reciprocals of the squares of the cube roots of positive natural numbers and two definite integrals. By using the differences between the sums and the integrals are formed in a natural way two sequences of real numbers converging to the same finite limit. We completely determine the monotonicity character of the sequences which are obtained by convex combinations of the two sequences, in an elegant way. We also present an interesting result about monotonicity character of a real function on the interval <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13660_2025_3375_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="MATHML"><math> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mo>+</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$[1,+\infty )$</EquationSource> </InlineEquation>, which is used in the proof of the main result.</p>

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Monotonicity character of convex combinations of two sequences generated by the square of the cube root function

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月15日

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  • Stevo Stević,
  • Bratislav Iričanin,
  • Witold Kosmala,
  • Zdeněk Šmarda

摘要

There is a natural connection between the finite sums of the reciprocals of the squares of the cube roots of positive natural numbers and two definite integrals. By using the differences between the sums and the integrals are formed in a natural way two sequences of real numbers converging to the same finite limit. We completely determine the monotonicity character of the sequences which are obtained by convex combinations of the two sequences, in an elegant way. We also present an interesting result about monotonicity character of a real function on the interval [ 1 , + ∞ ) $[1,+\infty )$ , which is used in the proof of the main result.