Questions: Use the nth-term test for divergence to show that the series is divergent, or state that the test is inconclusive. The series is ∑ from n=1 to ∞ of n/(n+5). A. The test is inconclusive. B. The series diverges.

Solution Steps

To determine if the series diverges using the nth-term test for divergence, we need to evaluate the limit of the nth term as \( n \) approaches infinity. If the limit is not zero, the series diverges. If the limit is zero, the test is inconclusive.

We start with the nth term of the series given by

\[ a_n = \frac{n}{n + 5}. \]

Next, we calculate the limit of the nth term as \( n \) approaches infinity:

\[ \lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{n}{n + 5}. \]

By dividing the numerator and the denominator by \( n \), we have:

\[ \lim_{n \to \infty} \frac{1}{1 + \frac{5}{n}}. \]

As \( n \) approaches infinity, \( \frac{5}{n} \) approaches \( 0 \). Therefore, the limit simplifies to:

\[ \lim_{n \to \infty} a_n = \frac{1}{1 + 0} = 1. \]

According to the nth-term test for divergence, if

\[ \lim_{n \to \infty} a_n \neq 0, \]

the series diverges. Since we found that

\[ \lim_{n \to \infty} a_n = 1 \neq 0, \]

we conclude that the series diverges.

Final Answer