Question 1198622: Is the following series, 5 -10/3 +20/9 -40/27.
Convergent, or divergent?One of the following is the correct answer. Which one?
A) divergent
B) convergent
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Answer: Convergent
Reason:
The jump from 5 to -10/3 is "times -2/3"
The jump from -10/3 to 20/9 is "times -2/3"
The jump from 20/9 to -40/27 is "times -2/3"
We see that this sequence is geometric with common ratio r = -2/3
Because -1 < r < 1, this particular infinite geometric series is convergent.
Extra info:
To compute the value of 5 - 10/3 + 20/9 - 40/27 + ...
we'll use this formula
S = a/(1-r)
where in this case, a = 5 is the first term.
So,
S = a/(1-r)
S = 5/(1-(-2/3))
S = 3
The infinitely many terms add up to 3.
Realistically we can only add up finitely many terms, which means the sum will steadily approach 3 but never actually arrive at this exact value.
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