Q. 3 - Calculus

Question

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

Step-by-Step Solution

Answer

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

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

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, $\left|a_k-L\right|<\in for k\ge N$.

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