Chapter 7: Q. 4 (page 614)
What is meant by the tail of a series? Why does it make sense that the tail of a series determines the convergence of the series?
Short Answer
For given , there exists a positive integer Nsuch that for .
By definition of convergence, only the terms after N that depends on the given determine the convergence. These are the terms that define the tail of the sequence.
Thus, the convergence of a series depends only on the convergence of the tail of the series.