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Conditional Convergence of Series

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Given the series S=∑n=1∞(−1)n+1nS = \displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}, which of the following series has the same convergence behavior as SS?

I. ∑n=1∞(−1)n+12n\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{2n}

II. ∑n=1∞(−1)n+1n+sin⁡(n)\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n + \sin(n)}

III. ∑n=1∞(−1)n+1nn3/2\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}\sqrt{n}}{n^{3/2}}

A

All three series\text{All three series}

B

I and II only\text{I and II only}

C

I and III only\text{I and III only}

D

I only\text{I only}

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