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

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

    \[ \sum_{n=1}^{\infty}  \frac{n \cos^2 \left( \frac{n \pi}{3} \right)}{2^n}. \]


The series converges. We have

    \[ \frac{n \cos^2 \left( \frac{n \pi}{3} \right)}{2^n} \leq \frac{n}{2^n} \qquad \text{for all }n. \]

But we know from a previous exercise (Section 10.14, Exercise #3) that

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

converges. Hence,

    \[ \sum_{n=1}^{\infty} \frac{n \cos^2 \left( \frac{n \pi}{3} \right)}{2^n} \]

converges as well.

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