Does the following infinite geometric series diverge or converge? Explain.
1/5 + 1/25 + 1/125 + 1/625
A) It diverges; it has a sum.
B) It converges; it has a sum.
C) It diverges; it does not have a sum.
D) It converges; it does not have a sum.
I am pretty sure that it is divergent. I am only confused by the part of it having a sum. How do I know when a geometric series has a sum.
I found this formula for it. -1>r>1
Sorry, I meant converge. I went back and looked at the video. They said converge.
Notice that is simply a geometric series, where
a = 1/5, r = 1/5
since Sum(all terms) = a/(1-r)
= (1/5)/(4/5)
= 1/4 , it clearly converges
What made you think it diverges?
What have you studied about diverging and converging series?
( simply adding the first 4 already gives us .2496 )
To determine whether an infinite geometric series converges or diverges, you need to examine the common ratio (r) of the series.
In this case, the series is given as 1/5 + 1/25 + 1/125 + 1/625.
To find the common ratio, divide any term by its previous term. In this series, the common ratio r is 1/5 ÷ 1/25, which simplifies to:
(1/5) ÷ (1/25) = (1/5) × (25/1) = 5/5 = 1.
When the common ratio (r) is between -1 and 1, the series converges and has a sum. When the common ratio is outside of this range, the series diverges and does not have a sum.
In this case, the common ratio is 1, which is greater than 1. Therefore, the series diverges, meaning it does not have a finite sum.
So, the correct answer is C) It diverges; it does not have a sum.
To determine whether an infinite geometric series converges or diverges, you need to look at the common ratio (r) -- the number you multiply by to get from one term to the next. In this case, the common ratio is 1/5 because every term is obtained by dividing the previous term by 5.
The formula to determine if an infinite geometric series converges or diverges is as follows:
- If the absolute value of the common ratio (|r|) is less than 1, then the series converges and has a sum.
- If the absolute value of the common ratio (|r|) is greater than or equal to 1, then the series diverges and does not have a sum.
In this case, |1/5| = 1/5, which is less than 1. Therefore, the series converges and has a sum.
To find the sum of a convergent geometric series, you can use the formula:
S = a / (1 - r)
Where:
- S is the sum of the series,
- a is the first term of the series, and
- r is the common ratio.
In this case, a = 1/5 and r = 1/5. Substituting these values into the formula, we get:
S = (1/5) / (1 - 1/5)
Simplifying the expression further:
S = (1/5) / (4/5)
S = 1/5 * 5/4
S = 1/4
So, the sum of the given infinite geometric series is 1/4.
Therefore, the correct answer is B) It converges; it has a sum.