Abstract
In this paper, we investigate the reciprocal sums of even and odd terms in the Fibonacci sequence, and we obtain four interesting families of identities which give the partial finite sums of the even-indexed (resp., odd-indexed) reciprocal Fibonacci numbers and the even-indexed (resp., odd-indexed) squared reciprocal Fibonacci numbers.
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1 Introduction
The Fibonacci sequence is defined by the linear recurrence relation
where \(F_{n}\) is called the nth 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 [1–5].
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 \(n\geq1\),
Wu and Zhang [11, 12] generalized these identities to the Fibonacci polynomials and Lucas polynomials, and they considered the subseries of infinite sums derived from the reciprocals of the Fibonacci polynomials and Lucas polynomials.
Recently, Wu and Wang [13] studied the partial finite sum of the reciprocal Fibonacci numbers and deduced the following main result.
Theorem 1.3
For all \(n\geq4\),
Inspired by Wu and Wang’s work, Wang and Wen [14] strengthened Theorem 1.1 and 1.2 to the finite sum case.
Theorem 1.4
If \(m\geq3\) and \(n\geq2\), then
Theorem 1.5
For all \(m\geq2\) and \(n\geq1\), we have
Applying elementary methods, we investigate the partial finite sums of the even-indexed and odd-indexed reciprocal Fibonacci numbers in this paper, and obtain four interesting families of identities. In Section 2, we consider the reciprocal sums of even and odd terms in the Fibonacci sequence. In Section 3, we present the finite sums of the even-indexed and odd-indexed squared reciprocal Fibonacci numbers.
2 Main results I: the reciprocal sums
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
If \(n\geq1\), then
The following is an interesting identity concerning the Fibonacci numbers.
Lemma 2.3
Assume that a and b are two integers with \(a\geq b\geq0\). If \(n>a\), then
Proof
We proceed by induction on n. It is clearly true for \(n=a+1\). Assuming the result holds for any integer \(n>a\), we show that the same is true for \(n+1\).
Applying (2.2) repeatedly and by the induction hypothesis, we get
which completes the induction on n. □
Remark
Recently, Akyiğit et al. [15, 16] defined the split Fibonacci quaternion, the split Lucas quaternion and the split generalized Fibonacci quaternion, and they obtained some similar identities to those above for these quaternions.
Before presenting our main results, we establish an inequality.
Proposition 2.4
If \(n\geq3\), then
Proof
A direct calculation shows that it is true for \(n=3\). Thus, we assume that \(n\geq4\) in the rest of the proof.
Setting \(a=2\) and \(b=0\), and replacing n by 2n in (2.6) yields
From (2.5), we know that
Applying (2.8), (2.9), and the fact \(F_{2n-3}\geq2\) and \(F_{2n-1}F_{2n}>F_{2n+1}\) if \(n\geq3\), we obtain
which is equivalent to
Now we have
It is not hard to see that for \(n\geq4\), \(F_{2n-3}\geq n+1\), which completes the proof. □
Now we introduce our main results on the reciprocal sums of Fibonacci numbers.
Theorem 2.5
For all \(n\geq3\), we have
Proof
By elementary manipulations and (2.1), we derive that, for \(k\geq1\),
Hence, we have
It follows from (2.4) that
which implies that
Invoking (2.1) again, we can readily deduce that
from which we obtain
Because of (2.7), we get, if \(n\geq3\),
Combining (2.11) and (2.13), we have
which yields the desired identity. □
Theorem 2.6
If \(m\geq3\) and \(n\geq1\), we have
Proof
It is obviously true for \(n=1\). Now we assume that \(n\geq2\).
By some calculations and (2.1), we obtain, for \(k\geq2\),
from which we have
On the other hand, it follows from (2.12) that
We claim that if \(n\geq1\) and \(m\geq3\),
Replacing a by \(a-1\) in (2.2), we arrive at
which implies that
Thus, \(F_{2n-1}F_{2n}F_{2n+1}\leq F_{6n}< F_{6n+1}\leq F_{2mn+1}\), which means
Combining (2.16) and (2.18) yields
from which the desired result follows immediately. □
Corollary 2.7
For all \(n\geq1\), we have
Proof
By using (2.15) repeatedly, we have
Thus, we obtain
Applying the same argument to (2.12) yields
Hence we have
which completes the proof. □
Remark
Identity (2.19) can be regarded as the limit of (2.14) as \(m\to\infty\).
Theorem 2.8
For all \(n\geq1\) and \(m\geq2\), we have
Proof
It is clearly true for \(n=1\), hence we suppose that \(n\geq 2\) in the following.
Invoking (2.1), we derive that for \(k\geq2\),
which implies that
It follows from (2.17) that
based on which we conclude that, when \(n>1\),
Employing (2.1) again, we can readily obtain
from which we arrive at
Combining the above inequality with (2.21), we have
which yields the desired result. □
As m approaches infinity, Theorem 2.8 becomes the following.
Corollary 2.9
If \(n\geq1\), we have
3 Main results II: the reciprocal square sums
We first introduce several preliminary results on the square of the Fibonacci numbers.
Lemma 3.1
For all \(n\geq2\), we have
Proof
It follows from
that
where the last equality follows from (2.1). □
Lemma 3.2
If \(n\geq2\), then
Proof
It is straightforward to check that
where the last equality follows from (2.6). □
Lemma 3.3
For each \(n\geq2\), we have
Proof
A direct calculation shows that
The proof is complete. □
Remark
In fact, applying the equalities (ii) and (iv) of Proposition 2.2 of [17], we can easily obtain
where \(L_{n}\) means the nth Lucas number. Then (3.3) follows immediately from the fact \(L_{n}>F_{n}\) for \(n\geq2\).
Now we are ready to present the reciprocal square sums of the Fibonacci numbers.
Theorem 3.4
For all \(n\geq1\) and \(m\geq2\), we have
Proof
It is clearly true for \(n=1\), so we assume that \(n\geq2\) in the rest of the proof.
For \(k\geq2\), we have
It follows from (2.3) that
As a consequence of (3.1), we see
Applying (2.1), (3.5), (3.6), and (3.7), we derive that
which implies that
Thus, we have
Employing the same argument as above, we obtain, for \(k\geq2\),
For each \(k\geq2\), we have
Therefore,
from which we arrive at
Combining (3.8) and (3.9) yields
from which the desired result follows. □
As m tends to infinity in Theorem 3.4, we have the following consequence.
Corollary 3.5
For all \(n\geq1\), we have
Theorem 3.6
If \(n\geq1\) and \(m\geq2\), then
Proof
It is obvious when \(n=1\), thus we assume that \(n\geq2\) in the following.
It follows from (2.3) that
Therefore, applying (3.2), we deduce
For \(k\geq2\), we have
from which we derive
Employing (3.2) and (3.12), we obtain
where the last inequality follows from (3.3).
Now we see that, for \(k\geq2\),
which implies that
It is easy to see that
Hence,
It follows from (3.13) and (3.14) that
which completes the proof. □
Consequently, we have the following result.
Corollary 3.7
If \(n\geq1\), then
4 Conclusions
In this paper, we give the exact integral values of the reciprocal sums (resp., square sums) of the even and odd terms in the Fibonacci sequence. The results are new and important for those with closely related research interests. In addition, the methods used here are very elementary and can be extended to the investigation of other combinatorial sequences.
In a future paper, the reciprocal sums and the reciprocal square sums of the Fibonacci 3-subsequences will be presented.
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Acknowledgements
The authors would like to thank the anonymous referees for their helpful suggestions and comments which improved significantly the presentation of the paper. This work was supported by the National Natural Science Foundation of China (No. 11401080).
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Wang, A.Y., Zhang, F. The reciprocal sums of even and odd terms in the Fibonacci sequence. J Inequal Appl 2015, 376 (2015). https://doi.org/10.1186/s13660-015-0902-2
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DOI: https://doi.org/10.1186/s13660-015-0902-2