Which of the following statements is true for the series the summation from k=1 to infinity of the sin(k)/k^2? (10 points)

A) The series converges conditionally but not absolutely.
B) The series is divergent.
C) The series cannot be tested because it does not start at k = 0.
D) The series converges absolutely.

A 1/k^2 converges, and |sin(k)| < 1, so ...

To determine whether the series converges conditionally, converges absolutely, or diverges, we can use the concept of convergence of series.

The given series is the summation from k = 1 to infinity of sin(k)/k^2.

To test the convergence, we can apply the Comparison Test. Here's how to do it:

Step 1: Determine whether the terms of the series are positive or negative.
In this case, the term sin(k)/k^2 is always positive because sin(k) is always positive or zero, and k^2 is also positive or zero.

Step 2: Choose a series that is known to converge or diverge and compare it with the given series.
To apply the Comparison Test, we need to find a comparison series whose behavior is already known.

In this case, we can use the comparison series with the term 1/k^2, which is the harmonic series with a p-value of 2.

Comparing the given series sin(k)/k^2 with the series 1/k^2, we have

0 ≤ sin(k)/k^2 ≤ 1/k^2 for all k ≥ 1

Since the harmonic series 1/k^2 converges (p-value is 2 > 1), the given series sin(k)/k^2 is also convergent by the Comparison Test.

Step 3: Determine whether the series converges conditionally, converges absolutely, or diverges.
Since the given series sin(k)/k^2 is convergent, we need to determine whether it converges absolutely or conditionally.

For a series to converge absolutely, the absolute value of each term must converge. In this case, the absolute value of sin(k)/k^2 is equal to |sin(k)/k^2| = sin(k)/k^2, which is the same as the original series.

Therefore, the given series sin(k)/k^2 converges absolutely because the original series converges.

So, the correct statement is:

D) The series converges absolutely.