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

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): You can put this solution on YOUR website!

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.

RELATED QUESTIONS
For the infinite geometric series below, identify whether it converges or diverges.... (answered by greenestamps,math_tutor2020)

Determine whether the sequence converges or diverges. If it converges, give the limit.... (answered by Boreal)

please help me solve the following: determine whether the infinite geometric series... (answered by ewatrrr,MathLover1)

Does the infinite geometric series diverge or converge? Explain. 2+6+18+54+... a -... (answered by Edwin McCravy)

Please help me solve this problem. It says, Decide whether each infinite geometric... (answered by jim_thompson5910)

Determine whether the infinite geometric series converges. If it does, then find the sum (answered by jim_thompson5910)

Determine whether the sequence converges or diverges. If it converges, give the limit.... (answered by MathLover1)

I need help on these questions as I'm preparing for a unit test. Please help Is the... (answered by Alan3354)

Determine whether the sequence converges or diverges. If it converges, give the limit.... (answered by MathLover1)