\( S_{1}=\sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}} \) \( S_{2}=\sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) Which series converges?

We consider the series one by one. For S₂:   S₂ = ∑ₙ₌₁∞ 1/√n is a p-series with p = 1/2, which is less than 1. A p-series converges only if p > 1. Therefore, the series S₂ diverges. For S₁:   S₁ = ∑ₙ₌₁∞ (-1)ⁿ/√n is an alternating series. The Alternating Series Test tells us that if the absolute value of the terms decreases monotonically to 0, then the alternating series converges. In this case, the absolute value of the terms is 1/√n, which is indeed monotonically decreasing and tends to 0 as n → ∞. Thus, S₁ converges by the Alternating Series Test. Conclusion:   S₁ converges (conditionally), while S₂ diverges.