for the geometric series shown, state whether the series in convergent. If the series is convergent, give the sum
9 +6.3 + 4.41
It is convergent and the sum is
9(1 + 0.7 + 0.7^2 + ...) = 9/(1-0.7)= 30
where'd the 0.7 come from?
0.7 is the ratio of successive terms. You can verify that for yourself
To determine whether a geometric series is convergent, we need to check the common ratio (r) of the series. The common ratio is found by dividing any term in the series by its previous term.
In this case, let's find the common ratio (r):
r = 6.3 / 9 = 4.41 / 6.3
r = 0.7
To determine if the geometric series is convergent or not, we need to check the absolute value of r. If the absolute value of r is less than 1, then the series is convergent. Otherwise, if the absolute value of r is greater than or equal to 1, then the series is divergent.
In this case, the absolute value of r, |0.7|, is less than 1. Therefore, the series is convergent.
To find the sum of a convergent geometric series, we can use the formula:
sum = a / (1 - r)
where a is the first term of the series and r is the common ratio.
Let's calculate the sum using this formula:
sum = 9 / (1 - 0.7)
sum = 9 / 0.3
sum = 30
Therefore, the sum of the given geometric series is 30.