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Determine the convergence of the series (-1)(n(n-1))/2 / 2n

Consider the series

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

Determine whether the series converges or diverges. If it converges, determine whether it converges conditionally or absolutely.


The given series is absolutely convergent.

Proof. The convergence is absolute since

    \[ \sum_{n=1}^{\infty} \left| \frac{(-1)^{\frac{n(n-1)}{2}}}{2^n} \right| = \sum_{n=1}^{\infty} \frac{1}{2^n}, \]

which is an absolutely convergent geometric series. \qquad \blacksquare

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