@article{Li2025, 
author = {Hongjian Li and Kaili Yang and Pingzhi Yuan},
title = {The asymptotic behavior of the reciprocal sum of generalized Fibonacci numbers},
year = {2025},
journal = {Electronic Research Archive},
volume = {33},
number = {1},
pages = {409-432},
keywords = {generalized Fibonacci number, reciprocal sum, asymptotic formulas},
url = {https://www.sciopen.com/article/10.3934/era.2025020},
doi = {10.3934/era.2025020},
abstract = {Let              (              u        n            )              n      ≥      0       be the special Lucas    u-sequence defined by         u          n      +      2        =  A      u          n      +      1        −  B      u    n    ,        u    0    =  0  ,        u    1    =  1  ,where    n  ≥  0,    B  =  ±  1, and    A is an integer such that        A    2    −  4  B  >  0. Let         a    k    =      1          u              m        k            s        ,        1                  u                  m          k                    +              u                  m          k          +          l                      ,        1                  ∑                  i          =          0                l                    u                  m          k          +          i                      ,        1                  u                  m          k                            u                  m          k          +          2          l                      ,        1                  u                  m          k                            u                  m          k          +          2          l          −          1                      ,        1                  u                  m          k                    +      C        ,where    m  ,    l are positive integers,    s  =  1  ,  2  ,  3  ,  4, and    C is any constant. The aim of this paper is to find a form        g    n   such that         lim          n      →      ∞            (                            (                                    ∑                              k                =                n                            ∞                                      a              k                                )                          −          1                    −              g        n              )    =  0.For example, we show that         lim          n      →      ∞            (                            (                                    ∑                              k                =                n                            ∞                                      1                              u                                  m                  k                                                              )                          −          1                    −              (                              u                          m              n                                −                      u                          m              (              n              −              1              )                                      )              )    =  0.}
}