Finding the Interval of Convergence for the Series ∑(x+5)^n/4^n
When dealing with infinite series, it is crucial to determine the range of values for the variable x where the series converges. This range is known as the interval of convergence. In this article, we will explore the steps to find the interval of convergence for the series:
∑(x+5)^n/4^n
Step 1: Identify the Series Type
The given series is a power series, where the general term is of the form (x+5)^n/4^n. Power series are a special type of infinite series that are particularly important in mathematical analysis.
Step 2: Apply the Ratio Test
The ratio test is a powerful convergence test that can be used to determine the interval of convergence for power series. The ratio test states that if the limit of the ratio of consecutive terms of a series exists and is less than 1, then the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.
For the given series, the ratio of consecutive terms is:
(x+5)^(n+1)/4^(n+1) / (x+5)^n/4^n = (x+5)/4
The limit of this ratio as n approaches infinity is:
lim(n->∞) (x+5)/4 = (x+5)/4
Now, we can apply the ratio test:
- If |(x+5)/4| < 1, then the series converges.
- If |(x+5)/4| > 1, then the series diverges.
- If |(x+5)/4| = 1, then the test is inconclusive.
Solving the inequality |(x+5)/4| < 1, we get:
-5 < x < 3
Step 3: Determine the Interval of Convergence
The interval of convergence is the set of all values of x for which the series converges. Based on the results from the ratio test, the interval of convergence for the series ∑(x+5)^n/4^n is:
-5 < x < 3
This means that the series converges for all values of x strictly between -5 and 3, and diverges for all values of x outside this interval.
It's important to note that the endpoints of the interval of convergence (-5 and 3) are not included in the interval, as the ratio test is inconclusive at these points. To determine the behavior of the series at the endpoints, you would need to apply additional tests, such as the direct comparison test or the limit comparison test.
In summary, the interval of convergence for the series ∑(x+5)^n/4^n is the open interval (-5, 3).