Use the divergence test to determine if the following series diverge or state that the test is inconclusive. 
${\sum }_{n=1}^{\infty }\sin \left(\frac{n\pi }{2}\right)$

• A. Divergent
• B. Convergent
• C. Inconclusive

Answer: (A)

Verified step by step guidance

1. Step 1: Recall the divergence test (also known as the nth-term test for divergence). This test states that if the limit of the nth term of a series, a_n, as n approaches infinity does not equal 0, or if the limit does not exist, then the series diverges. If the limit equals 0, the test is inconclusive.
2. Step 2: Identify the nth term of the given series. Here, the nth term is a_n = sin(nπ/2).
3. Step 3: Analyze the behavior of sin(nπ/2) as n increases. Notice that the sine function oscillates periodically. Specifically, sin(nπ/2) takes on the values 0, 1, 0, -1, and repeats in this pattern as n increases.
4. Step 4: Since sin(nπ/2) does not settle to a single value as n approaches infinity (it oscillates), the limit of a_n = sin(nπ/2) does not exist.
5. Step 5: By the divergence test, since the limit of a_n does not exist, the series ∑_{n=1}^{∞} sin(nπ/2) diverges.