Answer:
d. Diverges; the series is a constant multiple of the harmonic series.
Step-by-step explanation:
Given
[tex]Series: \frac{1}{6} + \frac{1}{12} + \frac{1}{18} + \frac{1}{24} + \frac{1}{30} + ...[/tex]
Required
Determine if it diverges or converges
From the series, we have:
[tex]T_1 = \frac{1}{6} = \frac{1}{6 * 1}[/tex]
[tex]T_2 = \frac{1}{12} = \frac{1}{6 * 2}[/tex]
[tex]T_3 = \frac{1}{18} = \frac{1}{6 * 3}[/tex]
[tex]T_4 = \frac{1}{24} = \frac{1}{6 * 4}[/tex]
[tex]T_5 = \frac{1}{30} = \frac{1}{6 * 5}[/tex]
So, each term can be written as:
[tex]T_n = \frac{1}{6n}[/tex]
And the series is:
[tex]\limits^\infty_1\sum T_n = \limits^\infty_1\sum \frac{1}{6n}[/tex]
[tex]\limits^\infty_1\sum T_n = \frac{1}{6} * \limits^\infty_1\sum \frac{1}{n}[/tex]
[tex]\limits^\infty_1\sum T_n = \frac{1}{6} * \infty[/tex]
[tex]\limits^\infty_1\sum T_n = \infty[/tex]
Recall that:
[tex]\limits^\infty_1\sum \frac{1}{n}[/tex] is known as the harmonic series, and it diverges to infinity
Hence: (d) is correct
