Question

Explain why the convergence of a sequence depends only on the convergence of the tail of the sequence.

Short answer

For any $\in >0$, there exists some N > 0 such that if k > N, $\left|a_k-L\right|<\in fork\ge N$.

By definition, only the $\in$terms determines the convergence of the sequence. And this refers to the tail of the sequence.

Step-by-step solution

Step 1. Given Information.

The convergence of the sequence.

Step 2. The definition.

For any $\in >0$, there exists some N > 0 such that if k > N, $\left|a_k-L\right|<\in fork\ge N$.

By definition, only the $\in$terms determines the convergence of the sequence. And this refers to the tail of the sequence.