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I wish to determine whether or not the following infinite series converges:

$$\sum_{n=0}^{\infty} \frac{\mathrm{sin}\left(\big(n+\frac{1}{2}\big) \pi\right)}{1 + \sqrt{n}}$$

The problem is in the section discussing the alternating series test, although the problem set merely asks whether the series converges or diverges.

I do not believe we can use the alternating series test because it is not true that the sequence of the terms is nonincreasing, which is primarily due to the oscillation from the range of $\mathrm{sin}\big(n + \frac{1}{2}\big)$.

Any help would be appreciated.

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  • $\begingroup$ It's worth noting that even if the $\pi$ is outside the sine, Dirichlet's test still show convergence $\endgroup$ – Milo Brandt Feb 21 '16 at 1:20
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Hint:

$$\sin\left(\left(n+\frac12\right)\pi\right) = \cos(n\pi) = (-1)^n$$

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  • $\begingroup$ Ah, I see that I was treating the $\pi$ as a constant of the series rather than as a parameter of $\mathrm{sine}$. $\endgroup$ – Jeremiah Dunivin Feb 21 '16 at 0:46

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