Question

Which geometric series converges?

A. 1/2 + 1/4 + 1/8 + 1/16 + ...

B. 1/2 + 1 + 2 + 4 + ...

C. 1/2 + 3/2+ 9/2+ 27/2+...

D. 1/2 + 3 + 18 + 108 + ...

Answer 1

The answer is A. because

A.
$0.5 > 0.25 > 0.125 > 0.06125$
B.
$0.5 < 1 < 2 < 4$
C.
$0.5 < 1.5 < 4.5 < 13.5$
D.
$0.5 < 3 < 8 < 108$

Answer 2

Answer:

Option: A is the correct answer.

          A)      $\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+.........$

Step-by-step explanation:

We know that a geometric series of the type:

$a_1+a_2+a_3+a_4+...$

where $a_1=a\\\\a_2=ar\\\\a_3=ar^2\\.\\.\\.\\.\\.\\.$

converges if r<1

Hence, we will check in each of the given options for which r<1

B)

 $\dfrac{1}{2}+1+2+4+...$

From this series we observe that the common ratio i.e. r=2

Since,

$a_1=\dfrac{1}{2}\\\\a_2=\dfrac{1}{2}\times 2\\\\\\a_2=1\\\\a_3=1\times 2=2\\\\a_4=2\times 2=4$

and so on.

Hence, the series is not convergent.

Hence, option: B is incorrect.

C)

 $\dfrac{1}{2}+\dfrac{3}{2}+\dfrac{9}{2}+\dfrac{27}{2}+....$

From this series we observe that the common ratio i.e. r=3>1

Hence, the series diverges.

Hence, option: C is false.

D)

 $\dfrac{1}{2}+3+18+108+....$

In this geometric series trhe common ratio is: 6>1

Hence, the series does not converge.

Hence, option D is incorrect.

A)

  $\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+.........$

Here we have:

         common ratio i.e. r=1/2<1

Hence, the geometric series converge and the sum is given by:

$S=\dfrac{a}{1-r}\\\\\\S=\dfrac{\dfrac{1}{2}}{1-\dfrac{1}{2}}\\\\\\S=\dfrac{\dfrac{1}{2}}{\dfrac{1}{2}}=1$

         Hence, option: A is correct.