Question

question
determine if the sum of the following infinite series is defined.
9 + \frac{19}{2}+10 + \frac{21}{2}+...
select the correct answer below:
the sum is defined.
the sum is not defined.
it is impossible to tell if the sum is defined.

Explanation:

Step1: Identify the series type

This is an infinite series with terms \(a_1 = 9,a_2=\frac{19}{2},a_3 = 10,a_4=\frac{21}{2},\cdots\).

Step2: Check for convergence - divergence

The \(n\) - th term test for divergence: \(\lim_{n
ightarrow\infty}a_n\).
Rewrite the terms as \(a_n\): The terms do not approach \(0\) as \(n
ightarrow\infty\). In fact, the terms keep growing. For an infinite series \(\sum_{n = 1}^{\infty}a_n\), if \(\lim_{n
ightarrow\infty}a_n
eq0\), then the series diverges.
Since the series diverges, the sum of the infinite series is not defined.

Answer:

The sum is not defined.