Question

Look at the pic below for question :)

Answer 1

Answer:

OPTION B

Step-by-step explanation:

Geometric series : $\displaystyle \sum_{n = 0}^{ \infty} ar^n$

where $a$ is the first term of the series and

$r$ is common difference.

A geometric series is convergent if |r| < 1.

It is divergent otherwise.

Since the first term of the series is a and the second term is ar, the ration of second term and first term, $\frac{ar}{r}$ = r.

OPTION A:

$\frac{1}{81} + \frac{1}{27} + \frac{1}{9} + \frac{1}{3} + \hdots$.

Here, $a = \frac{1}{81}$ and $ar = \frac{1}{27}$

$\implies r= \frac{27}{81} = 3$

r > 1. So, this series is divergent.

OPTION B:

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \hdots$

a = 1; ar = $\frac{1}{2}$.

$\implies r = \frac{1}{2}$.

Since, r < 1, we can say that the series is convergent.

OPTION C:

We can easily see that |r| =4. So, it is not convergent.

OPTION D:

Again |r| = 2. So, the series should be divergent.