Editing Periodic sequence (section)

==Generalizations==
A sequence is '''eventually periodic''' or '''ultimately periodic'''<ref name=":0" /> if it can be made periodic by dropping some finite number of terms from the beginning. Equivalently, the last condition can be stated as  <math>a_{k+r} = a_k</math> for some ''r'' and sufficiently large ''k''. For example, the sequence of digits in the decimal expansion of 1/56 is eventually periodic:

: 1 / 56 = 0 . 0 1 7&nbsp;&nbsp;8 5 7 1 4 2&nbsp;&nbsp;8 5 7 1 4 2&nbsp;&nbsp;8 5 7 1 4 2&nbsp;&nbsp;...

A sequence is '''asymptotically periodic''' if its terms approach those of a periodic sequence.  That is, the sequence ''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;''x''<sub>3</sub>,&nbsp;... is asymptotically periodic if there exists a periodic sequence ''a''<sub>1</sub>,&nbsp;''a''<sub>2</sub>,&nbsp;''a''<sub>3</sub>,&nbsp;... for which

:<math>\lim_{n\rightarrow\infty} x_n - a_n = 0.</math><ref name=":2" />

For example, the sequence

:1 / 3,&nbsp;&nbsp;2 / 3,&nbsp;&nbsp;1 / 4,&nbsp;&nbsp;3 / 4,&nbsp;&nbsp;1 / 5,&nbsp;&nbsp;4 / 5,&nbsp;&nbsp;...

is asymptotically periodic, since its terms approach those of the periodic sequence 0, 1, 0, 1, 0, 1, ....