Question

Let f be the function defined by f(x)=3/(2x²−7x+5).

(d) Determine whether the series
∑{n=5,[infinity]} 3/(2n²−7n+5)
converges or diverges. State the conditions of the test used for determining convergence or divergence.

Answer 1

To determine whether the series converges or diverges, we can use the Comparison Test with a known convergent series. By comparing the given series to a convergent series, we can show that the given series also converges.

Step-by-step explanation:

To determine whether the series ∑n=5∞ 3/(2n²−7n+5) converges or diverges, we can use the Comparison Test. First, let's compare the given series with a known convergent or divergent series.

We know that the series ∑n=1∞ 1/n2 converges, so we will compare our series to this series.

Now, we need to find a positive constant M such that 3/(2n²−7n+5) ≤ M/(n2). Since the quadratic term dominates, we can choose M=3/2, and

3/(2n²−7n+5) ≤ 3/2n2 for n ≥ 1.

Since ∑ 1/n2 converges, the given series ∑n=5∞ 3/(2n²−7n+5) also converges by the Comparison Test.