# On the Sum of Reciprocal of Polynomial Applied to Higher Order Recurrences

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

## 2. Proof of the Theorem

## 3. Computational Aspects

**Example**

**1.**

**Example**

**2.**

## 4. A Generalization for Higher Dimensional Recurrences

**Theorem**

**2.**

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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${\mathit{u}}_{\mathit{n}}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\left|\mathit{\nu}\right|}^{\mathit{n}-1}$ |
---|---|---|---|

${u}_{n}=2{u}_{n-1}+7{u}_{n-2}+{u}_{n-3}+{u}_{n-4}$ | $3.8851$ | $1.7845$ | $1.8847\times {10}^{-17}$ |

${u}_{n}=2{u}_{n-1}+7{u}_{n-2}+5{u}_{n-3}+{u}_{n-4}$ | $4.0489$ | $-1$ | $8.6442\times {10}^{-31}$ |

n | ${\mathit{A}}_{\mathit{n}}$ | ${\mathit{B}}_{\mathit{n}}$ | ${\mathit{A}}_{\mathit{n}}/{\mathit{B}}_{\mathit{n}}$ | ${\left|\mathit{\nu}\right|}^{\mathit{n}-1}$ |
---|---|---|---|---|

10 | ≈66,125.65 | $65,940$ | $1.0028$ | $0.01916$ |

100 | $\approx 4.5\times {10}^{66}$ | $\approx 4.5\times {10}^{66}$ | $1+3\times {10}^{-34}$ | $6.6\times {10}^{-18}$ |

500 | $\approx 2.9\times {10}^{341}$ | $\approx 2.9\times {10}^{341}$ | $1+{10}^{-100}$ | $1.3\times {10}^{-86}$ |

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Trojovský, P.
On the Sum of Reciprocal of Polynomial Applied to Higher Order Recurrences. *Mathematics* **2019**, *7*, 638.
https://doi.org/10.3390/math7070638

**AMA Style**

Trojovský P.
On the Sum of Reciprocal of Polynomial Applied to Higher Order Recurrences. *Mathematics*. 2019; 7(7):638.
https://doi.org/10.3390/math7070638

**Chicago/Turabian Style**

Trojovský, Pavel.
2019. "On the Sum of Reciprocal of Polynomial Applied to Higher Order Recurrences" *Mathematics* 7, no. 7: 638.
https://doi.org/10.3390/math7070638