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**An identity related to Jordan’s inequality.**
*(English)*
Zbl 1143.26010

The paper establishes an identity that expresses \(\frac {\sin(x)} x\) as an infinite series of \((\pi^2 -4x^2)\). The partial sums of the series are then used to estimate \(\frac {\sin(x)} x\), which generalize the left hand side of Jordan’s inequality
\[
\frac 2 \pi \leq \frac {\sin(x)} x < 1, \qquad x \in (0, \pi / 2].
\]
The estimates are stated stronger than existing results. The paper also uses the Taylor expansion of \(\frac x {\sin(x)}\) to generalize the right hand side of Jordan’s inequality. The reciprocal of a partial sum of this series is used as an estimate on the right hand side.

Reviewer: Tieling Chen (Aiken)

### MSC:

26D15 | Inequalities for sums, series and integrals |

### Keywords:

Taylor expansion
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\textit{J.-L. Li}, Int. J. Math. Math. Sci. 2006, No. 18, 76782, 6 p. (2006; Zbl 1143.26010)

### References:

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