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

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

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

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

So, this geometric series will not converge.
Option (D) is incorrect.
Thus, the correct option is (A).