Question

Consider the following series.

Answer 1

Answer:

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.