Abstract
Let {un} be a higher-order recursive sequence. Several identities are obtained for the infinite sums and finite sums of the reciprocals of higher-order recursive sequences. MSC: Primary 11B39
Highlights
The so-called Fibonacci zeta function and Lucas zeta function defined by∞ ζF (s) = n= Fns∞ and ζL(s) = n= Lsn, where the Fn and Ln denote the Fibonacci numbers and Lucas numbers, have been considered in several different ways
2 Several lemmas To complete the proof of our theorem, we need the following
Since h = max{α– βn + n, d–n} < , there exists n ≥ n sufficiently large so that the modulus of the last error term becomes less than / , which completes the proof
Summary
Some authors considered the nearest integer of the sums of reciprocal Fibonacci numbers and other famous sequences and obtained several new interesting identities, see [ ] and [ ]. For any positive real number β > , there exists a positive integer n such that βn k=n uk Taking β → +∞, from Theorem we may immediately deduce the following.
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