Absolute Convergence Test

ACT (Absolute Convergence Test)

If $\sum | a_{n} |$ converges, then $\sum a_{n}$ converges.

Proof
Let $ε > 0$ . Since $\sum | a_{n} |$ converges, then we can choose $N \in N$ such that any $n > m \geq N$ implies:

| | a_{m + 1} | + \dots + | a_{n} | | < ε

Choose any $n > m \geq N$ . Then:

\begin{aligned} | a_{m + 1} + \dots + a_{n} | & \leq | a_{m + 1} | + \dots + | a_{n} | & Δ Inequality \\ = | | a_{m + 1} | + \dots + | a_{n} | | \\ < ε \end{aligned}

Thus the $\sum a_{n}$ is Cauchy, so then $\sum a_{n}$ converges.
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