Question

Look at the pic below for question :)

Answer 1

OPTION B

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,
`(ar)/(r)` = r.

OPTION A:

`$ (1)/(81) + (1)/(27) + (1)/(9) + (1)/(3) + \hdots $`.

Here,
`$ a = (1)/(81) $` and
`$ ar = (1)/(27) $`

`$ \implies r=  (27)/(81) = 3 $`

r > 1. So, this series is divergent.

OPTION B:

`$ 1 + (1)/(2) + (1)/(4) + (1)/(8) + \hdots $`

a = 1; ar =
`$ (1)/(2) $`.

`$  \implies r = (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.