Question

Which geometric series converges?
A. 2+0.2+0.02+0.002+...
B. 2+4+8+16+...
C. 2-20+200-2000+...
D. 2+2+2+2+...

Answer 1

Answer:  The correct option is (A) 2 + 0.2 + 0.02 + 0.002 +  .  .  .

Step-by-step explanation:  We are given to select the correct geometric series that converges.

We know that

a geometric series converges if the modulus of its common ratio is less than 1.

Option (A) : 2 + 0.2 + 0.02 + 0.002 +  .  .  .

Here, first term, a= 2  and the common ratio is given by

$r=\dfrac{0.2}{2}=\dfrac{0.02}{0.2}=\dfrac{0.002}{0.02}=~.~.~.~=0.1\\\\\Rightarrow |r|=|0.1|=0.1<1$

So, this geometric series will converge.

Option (A) is correct.

Option (B) : 2 + 4 + 8 + 16 +  .  .  .

Here, first term, a= 2  and the common ratio is given by

$r=\dfrac{4}{2}=\dfrac{8}{4}=\dfrac{16}{8}=~.~.~.~=2\\\\\Rightarrow |r|=|2|=2>1.$

So, this geometric series will not converge.

Option (B) is incorrect.

Option (C) : 2 - 20 + 200 - 2000 +  .  .  .

Here, first term, a= 2  and the common ratio is given by

$r=\dfrac{-20}{2}=\dfrac{200}{-20}=\dfrac{-2000}{200}=~.~.~.~=-10\\\\\Rightarrow |r|=|-10|=10>1.$

So, this geometric series will not converge.

Option (C) is incorrect.

Option (D) : 2 +2 + 2 + 2 +  .  .  .

Here, first term, a= 2  and the common ratio is given by

$r=\dfrac{2}{2}=\dfrac{2}{2}=\dfrac{2}{2}=~.~.~.~=1\\\\\Rightarrow |r|=|1|=1.$

So, this geometric series will not converge.

Option (D) is incorrect.

Thus, the correct option is (A).