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Conclude if the given series converges or diverges and justify the conclusion

Test the following series for convergence or divergence. Justify the decision.

    \[ \sum_{n=1}^{\infty} \frac{2+(-1)^n}{2^n}. \]


This series converges by the comparison test since

    \[ \frac{2+(-1)^n}{2^n} \leq \frac{3}{2^n} \qquad \text{for all }n \]

and

    \[ \sum_{n=1}^{\infty} \frac{3}{2^n} = 3 \sum_{n=1}^{\infty} \frac{1}{2^n} \]

converges.

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