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# On the partial finite sums of the reciprocals of the Fibonacci numbers

*Journal of Inequalities and Applications*
**volume 2015**, Article number: 73 (2015)

## Abstract

In this article, we obtain two interesting families of partial finite sums of the reciprocals of the Fibonacci numbers, which substantially improve two recent results involving the reciprocal Fibonacci numbers. In addition, we present an alternative and elementary proof of a result of Wu and Wang.

## Introduction

The *Fibonacci sequence* [1], Sequence A000045 is defined by the linear recurrence relation

where \(F_{n}\) is the *n*th *Fibonacci number* with \(F_{0}=0\) and \(F_{1}=1\). There exists a simple and non-obvious formula for the Fibonacci numbers,

The Fibonacci sequence plays an important role in the theory and applications of mathematics, and its various properties have been investigated by many authors; see [2–5].

In recent years, there has been an increasing interest in studying the reciprocal sums of the Fibonacci numbers. For example, Elsner *et al.* [6–9] investigated the algebraic relations for reciprocal sums of the Fibonacci numbers. In [10], the partial infinite sums of the reciprocal Fibonacci numbers were studied by Ohtsuka and Nakamura. They established the following results, where \(\lfloor\cdot\rfloor\) denotes the floor function.

### Theorem 1.1

*For all*
\(n\geq2\),

### Theorem 1.2

*For each*
\(n\geq1\),

Further, Wu and Zhang [11, 12] generalized these identities to the Fibonacci polynomials and Lucas polynomials and various properties of such polynomials were obtained.

Recently, Holliday and Komatsu [13] considered the generalized Fibonacci numbers which are defined by

with \(G_{0}=0\) and \(G_{1}=1\), and *a* is a positive integer. They showed that

and

More recently, Wu and Wang [14] studied the partial finite sum of the reciprocal Fibonacci numbers and deduced that, for all \(n\geq4\),

Inspired by Wu and Wang’s work, we obtain two families of partial finite sums of the reciprocal Fibonacci numbers in this paper, which significantly improve Ohtsuka and Nakamura’s results, Theorems 1.1 and 1.2. In addition, we present an alternative proof of (1.5).

## Reciprocal sum of the Fibonacci numbers

We first present several well known results on Fibonacci numbers, which will be used throughout the article. The detailed proofs can be found in [5].

### Lemma 2.1

*Let*
\(n\geq1\), *we have*

*and*

*if*
*a*
*and*
*b*
*are positive integers*.

As a consequence of (2.2), we have the following result.

### Corollary 2.2

*For all*
\(n\geq1\), *we have*

It is easy to derive the following lemma and we leave the proof as a simple exercise.

### Lemma 2.3

*For each*
\(n\geq1\), *we have*

We now establish two inequalities on Fibonacci numbers which will be used later.

### Lemma 2.4

*If*
\(n\geq6\), *then*

### Proof

It is easy to see that

Since \(n\geq6\), \(F_{n-2}-2>1\). So

which completes the proof. □

### Lemma 2.5

*For each*
\(n\geq3\), *we have*

### Proof

Applying (2.2), we get

Thus

Employing (2.3), we have

which yields the desired equation (2.8). □

The following are some inequalities on the sum of reciprocal Fibonacci numbers.

### Proposition 2.6

*For all*
\(n\geq2\), *we have*

### Proof

For all \(k\geq2\),

Invoking (2.1), we obtain \(F_{k-1}^{2}-F_{k}F_{k-2}=(-1)^{k}\). Therefore,

Now we have

Because of (2.5), we have

Thus, we arrive at

This completes the proof. □

### Proposition 2.7

*Assume that*
\(m\geq2\). *Then*, *for all even integers*
\(n\geq4\), *we have*

### Proof

By elementary manipulations and (2.1), we deduce that

Hence, for \(n\geq3\), we have

Since *n* is even,

from which we conclude that

The proof is complete. □

### Proposition 2.8

*If*
\(n\geq5\)
*is odd*, *then*

### Proof

It is straightforward to check that the statement is true when \(n=5\).

Now we assume that \(n\geq7\). Since *n* is odd, we have

Applying (2.7) and (2.6) yields

Employing (2.11) and the above two inequalities, (2.12) follows immediately. □

### Proposition 2.9

*Let*
\(m\geq3\)
*be given*. *If*
\(n\geq3\)
*is odd*, *we have*

### Proof

It is easy to see that

thus (2.13) holds for \(n=3\). Now we assume that \(n\geq 5\).

Based on (2.11) and using the fact *n* is odd, we have

It is clear that

Since \(m\geq3\) and invoking (2.8), we obtain

which implies

Therefore, (2.13) also holds for \(n\geq5\). □

Now we state our main results on the sum of reciprocal Fibonacci numbers.

### Theorem 2.10

*For all*
\(n\geq4\), *we have*

### Proof

Combining (2.9), (2.10), and (2.12), we conclude that, for all \(n\geq4\),

from which (2.14) follows immediately. □

### Remark

Identity (2.14) was first conjectured by Professor Ohtsuka, the first author of [10]. Based on the formula of \(F_{n}\) and using analytic methods, Wu and Wang [14] presented a proof of (2.14). In contrast to Wu and Wang’s work, the techniques we use here are more elementary.

### Theorem 2.11

*If*
\(m\geq3\)
*and*
\(n\geq2\), *then*

### Proof

It is clear that

Combining (2.9) and (2.10), we find that, for all even integers \(n\geq4\),

Thus (2.16) and (2.17) show that, for all \(m\geq3\),

provided that \(n\geq2\) is even.

Next we aim to prove that, for \(m\geq3\) and all odd integers \(n\geq 3\),

If \(n=3\), we can readily see that

thus (2.18) holds for \(n=3\). So in the rest of the proof we assume that \(n\geq5\).

It is not hard to derive that, for all \(k\geq5\),

Hence, we get

Finally, combining (2.19) with (2.13) yields (2.18). □

### Remark

As \(m\rightarrow\infty\), (2.15) becomes (1.1). Hence our result, Theorem 2.11, substantially improves Theorem 1.1.

## Reciprocal square sum of the Fibonacci numbers

We first give several preliminary results which will be used in our later proofs.

### Lemma 3.1

*For all*
\(n\geq1\),

### Proof

It is easy to show that

Employing (2.1), the desired result follows. □

### Proposition 3.2

*Given an integer*
\(m\geq2\)
*and let*
\(n\geq3\)
*be odd*, *we have*

### Proof

It is straightforward to check that, for each \(k\geq2\),

where the last equality follows from (3.1).

Since *n* is odd, we have

If *m* is even, then

If *m* is odd, then

Thus, (3.2) always holds. □

### Proposition 3.3

*Let*
*n*
*be odd*, *then we have*

### Proof

Invoking (2.1), we can readily derive that

Now we have

It is obvious that \(2F_{k-1}F_{k}\geq F_{k-1}F_{k+1}\). From (2.1) and the fact that *n* is odd, we obtain

which implies that

from which we conclude that

The proof is complete. □

### Proposition 3.4

*Suppose that*
\(m\geq2\)
*and*
\(n>0\)
*is even*. *Then*

### Proof

Applying (2.1), we can rewrite \(F_{k}^{4}\) as

In addition,

Combining (3.5) and (3.6) yields

Therefore,

which completes the proof. □

### Proposition 3.5

*If*
\(n>0\)
*is even*, *then*

### Proof

Employing (2.1), we can deduce that

Hence, since *n* is even, we have

It is easy to see that

thus

We claim that

First, by (2.6), we have

It follows from (2.3), (2.4), and (2.5) that

which implies that

Thus we obtain

which yields the desired (3.7). □

Now we introduce our main result on the square sum of reciprocal Fibonacci numbers.

### Theorem 3.6

*For all*
\(n\geq1\)
*and*
\(m\geq2\), *we have*

### Proof

We first consider the case when *n* is odd. If \(n=1\), the result is clearly true. So we assume that \(n\geq3\).

It follows from (3.3) that

Employing (3.2) and (3.9) yields

which implies that, if \(n>0\) is odd, we have

We now consider the case where \(n>0\) is even. It follows from (3.7) that

Combining (3.4) and (3.10), we arrive at

from which we find that, if \(n>0\) is even,

This completes the proof. □

### Remark

Theorem 1.2 can be regarded as the limiting case as \(m\rightarrow\infty\) in (3.8).

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## Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 11401080) and the Research Project of Education Teaching Reform of University of Electronic Science and Technology of China (No. 2013XJYEL032). The authors would like to thank the referees for helpful comments leading to an improvement of an earlier version.

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## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to deriving all the results of this article, and read and approved the final manuscript.

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**Open Access** This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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### Cite this article

Wang, A.Y., Wen, P. On the partial finite sums of the reciprocals of the Fibonacci numbers.
*J Inequal Appl* **2015, **73 (2015). https://doi.org/10.1186/s13660-015-0595-6

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### MSC

- 11B39

### Keywords

- Fibonacci numbers
- partial sums
- reciprocal