Question

Consider the following series.

Answer 1

Answer:

d. Diverges; the series is a constant multiple of the harmonic series.

Step-by-step explanation:

Given

$Series: \frac{1}{6} + \frac{1}{12} + \frac{1}{18} + \frac{1}{24} + \frac{1}{30} + ...$

Required

Determine if it diverges or converges

From the series, we have:

$T_1 = \frac{1}{6} = \frac{1}{6 * 1}$

$T_2 = \frac{1}{12} = \frac{1}{6 * 2}$

$T_3 = \frac{1}{18} = \frac{1}{6 * 3}$

$T_4 = \frac{1}{24} = \frac{1}{6 * 4}$

$T_5 = \frac{1}{30} = \frac{1}{6 * 5}$

So, each term can be written as:

$T_n = \frac{1}{6n}$

And the series is:

$\limits^\infty_1\sum T_n = \limits^\infty_1\sum \frac{1}{6n}$

$\limits^\infty_1\sum T_n = \frac{1}{6} * \limits^\infty_1\sum \frac{1}{n}$

$\limits^\infty_1\sum T_n = \frac{1}{6} * \infty$

$\limits^\infty_1\sum T_n = \infty$

Recall that:

$\limits^\infty_1\sum \frac{1}{n}$ is known as the harmonic series, and it diverges to infinity

Hence: (d) is correct