The first two terms of geometry series are x + 4 and x²- 16
For which values of x will the series converge
Calculate the sum to infinity of the service in terms of x
If the sum to infinity of the series is 5,determine the value of x
To determine the values of x for which the series will converge, we need to find the common ratio of the geometry series.
The common ratio (r) of a geometry series can be found by dividing any term by its preceding term.
So, (x²- 16) / (x + 4) = r
We can simplify this equation by factoring x² - 16.
(x + 4)(x - 4) / (x + 4) = r
Canceling out the common factor of (x + 4) on both sides, we get:
x - 4 = r
For the series to converge, the absolute value of the common ratio (|r|) must be less than 1.
|x - 4| < 1
This inequality can be solved as:
-1 < x - 4 < 1
Adding 4 to all sides, we get:
3 < x < 5
Therefore, the values of x for which the series will converge are x is greater than 3 and less than 5.
To calculate the sum to infinity of the series, we can use the formula:
S = a / (1 - r)
In this case, the first term (a) of the series is x + 4 and the common ratio (r) is x - 4.
So, S = (x + 4) / (1 - (x - 4))
Simplifying this expression, we get:
S = (x + 4) / (5 - x)
If the sum to infinity of the series is 5, then we can set S equal to 5 and solve for x.
5 = (x + 4) / (5 - x)
Multiplying both sides by (5 - x), we get:
25 - 5x = x + 4
Combining like terms, we have:
6x = 21
Dividing both sides by 6, we get:
x = 3.5
Therefore, the value of x for which the sum to infinity of the series is 5 is x = 3.5.