Mathematics: 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 +

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Which geometric series converges? a. 1/2 + 1/4 + 1/8

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13.03.2022

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

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

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.

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

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

Hence, the series diverges.

Hence, option: C is false.

$\dfrac{1}{2}+3+18+108+....$

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

Hence, the series does not converge.

Hence, option D is incorrect.

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

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.

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