10. Which of the following is an interval of convergence for the series

Let,

Then,

Thus, by the Ratio Test, the given series is absolutely convergent and therefore convergent whenever,

Since the Ratio Test does not tell us the behavior of the given series at the endpoints, we need to test each end point by substituting into the given series as follows:

 The first substitution above yields a series that converges by the Alternating Series Test.

The second substitution above yields a harmonic series, which is divergent.

Thus the interval of convergence for the given series is,