Answered

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Consider the following series.

1/6 + 1/12 + 1/18 + 1/24 + 1/30

Determine whether the series is convergent or divergent. Justify your answer.

a. Converges; the series is a constant multiple geometric series.
b. Converges; the limit of the terms, an, is 0 as n goes to infinity.
c. Diverges; the limit of the terms, an, is not 0 as n goes to infinity.
d. Diverges; the series is a constant multiple of the harmonic series.

If it is convergent, find the sum.

Sagot :

MrRoyal

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

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