Question

use the nth term test for divergence to show that the series is divergent, or state that the test is inconclusive
sum _ { n = 0 } ^ { infty } \frac { 1 } { n + 14 }
select the correct choice below and, if necessary, fill in the answer box within your choice
a. the series diverges because ( lim _ { n
ightarrow infty } \frac { 1 } { n + 14 } ) exists and is equal to
b. the series diverges because ( lim _ { n
ightarrow infty } \frac { 1 } { n + 14 } = infty ) and fails to exist
c. the series diverges because ( lim _ { n
ightarrow infty } \frac { 1 } { n + 14 } = - infty ) and fails to exist
d. the test is inconclusive because ( lim _ { n
ightarrow infty } \frac { 1 } { n + 14 } = )

Explanation:

Step1: Find the limit of the general term

We need to find \(\lim_{n
ightarrow\infty}\frac{1}{n + 14}\).
Divide both the numerator and denominator by \(n\):
\(\lim_{n
ightarrow\infty}\frac{\frac{1}{n}}{1+\frac{14}{n}}\)

Step2: Evaluate the limit

As \(n
ightarrow\infty\), \(\lim_{n
ightarrow\infty}\frac{1}{n}=0\) and \(\lim_{n
ightarrow\infty}\frac{14}{n}=0\).
So, \(\lim_{n
ightarrow\infty}\frac{\frac{1}{n}}{1+\frac{14}{n}}=\frac{0}{1 + 0}=0\).
But according to the \(n\) - th term test for divergence: If \(\sum_{n = 0}^{\infty}a_{n}\) is a series and \(\lim_{n
ightarrow\infty}a_{n}
eq0\), the series diverges. If \(\lim_{n
ightarrow\infty}a_{n}=0\), the test is inconclusive.

Answer:

D. The test is inconclusive because \(\lim_{n
ightarrow\infty}\frac{1}{n + 14}=0\)