Test for Divergence (Series)

Test for Divergence

If $\sum a_{n}$ converges then $a_{n} \to 0$ .

Proof
Let $ε > 0$ . By the Cauchy Condition for series, then choose $N \in N$ such that $n > m \geq N$ implies:

| a_{m + 1} + \dots + a_{n} | < ε

In particular, use $m = n - 1$ and any $n > N$ then:

| a_{n} | = | a_{n} - 0 | < ε

So by definition then $a_{n} \to 0$ .
☐