Question

Consider the following series. 1 4 + 1 8 + 1 12 + 1 16 + 1 20 Determine whether the series is convergent or divergent. Justify your answer. Converges; the series is a constant multiple geometric series. Converges; the limit of the terms, an, is 0 as n goes to infinity. Diverges; the limit of the terms, an, is not 0 as n goes to infinity. Diverges; the series is a constant multiple of the harmonic series. Changed: Your submitted answer was incorrect. Your current answer has not been submitted.

Answer 1

Answer:

The series diverges

Step-by-step explanation:

First, let's find the sequence:

$4, 8, 12, 16, 20,...,$

As we can see, they are multiples of 4, so one possible sequence is:

$4,8,12,16,20,...,=4n\hspace{8};\hspace{3}n\in Z$

Hence, we can represent the series as:

$\sum_{n=1}^{\infty} 4n$

Let's use the zero test which states:

$\Sigma a_n \\Diverges\hspace{3}if\\\\ \lim_{n \to \infty} a_n \neq 0$

So:

$\lim_{n \to \infty} 4n= 4 \lim_{n \to \infty} n\\ \\ \lim_{n \to \infty} n=\infty\\ \\4\infty= \infty\\\\Hence\\\\ \lim_{n \to \infty} 4n=\infty$

Therefore, by the zero test, the series diverges