SOLUTION: For the infinite geometric series below, identify whether it converges or diverges. 10, 5, 2.5, 1.25, .... a. Converges b. Neither c. Diverges d. Both

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Question 1162453: For the infinite geometric series below, identify whether it converges or diverges.
10, 5, 2.5, 1.25, ....
a. Converges
b. Neither
c. Diverges
d. Both
Answer by jim_thompson5910(35256) (Show Source):
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Answer: A) converges

We can tell based on how the terms steadily get smaller and approach 0. As we add on more and more terms, the sum will get larger but not infinitely so. Instead the sum will slowly approach some finite value. The sum will never actually reach that value (since we can't reach infinity)

For the geometric sequence {10, 5, 2.5, 1.25, ...} the common ratio is 1/2 = 0.5
You can find this by dividing any term by its prior one
example:
common ratio = (second term)/(first term)
common ratio = 5/10
common ratio = 1/2
common ratio = 0.5

We multiply each term by 1/2 = 0.5 to get the next term. Since the common ratio r = 0.5 is between -1 and 1, the infinite series converges.

Put another way: if is true, then the infinite geometric series converges. Specifically it converges to the sum with 'a' being the first term. If either or , then the series diverges.