Mathematics: Consider the following series. 1/4 + 1/8 + 1+12 + 1/16 + 1/20 Required: Determine whether the geometric series is convergent or divergent.

Cause it didn t want to wing it...

Consider the following series. 1/4 + 1/8 + 1+12

Ответ:

keltaylor

13.06.2021

Mathematics

The series is convergent.

Step-by-step explanation:

1/4 + 1/8 + 1+12 + 1/16 + 1/20

In each term, the numerator stays 1, while the denominator is multiplied by 4. Thus, the series is given by:

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

Convergence test:

We compare the sequence of this test, $f_n = \frac{1}{4n}$ , with a sequence $g_n = \frac{1}{n}$ .

If $\lim_{n \rightarrow \infty} \frac{f_n}{g_n} \neq 0$ , the series is convergent. So

$\lim_{n \rightarrow \infty} \frac{f_n}{g_n} = \lim_{n \rightarrow \infty} \frac{\frac{1}{4n}}{\frac{1}{n}} = \lim_{n \rightarrow \infty} \frac{n}{4n} = \lim_{n \rightarrow \infty} \frac{1}{4} = \frac{1}{4} \neq 0$

As the limit is different of zero, the series is convergent.

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Consider the following series. 1/4 + 1/8 + 1+12 + 1/16 + 1/20

Required:
Determine whether the geometric series is convergent or divergent.

Ответ:

Ответ:

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