Find the radius of convergence and interval of convergence of the series.
n=2 series to infinity (-1)^n * x^n+6/n+7
R= ?
I= (or[ , ]or)
How do i do this?? ( means convergent and [ means divergent.
i know that the ratio test is used to figure it out. you first find the limits right?
To find the radius of convergence and interval of convergence of a power series, we can use the ratio test.
The ratio test states that if we have a series ∑ a_n(x - c)^n, then the series converges when the limit of the absolute value of the ratio of the consecutive terms, as n approaches infinity, is less than 1. It diverges when the limit is greater than 1 and the test is inconclusive when the limit is 1.
In this case, we have the series ∑ (-1)^n * x^(n+6)/(n+7).
Applying the ratio test, we compute the limit:
lim(n→∞) |[(-1)^(n+1) * x^(n+7)/(n+8)] / [(-1)^n * x^(n+6)/(n+7)]|
Simplifying, we get:
lim(n→∞) |-(n+7)/(n+8) * x|
Since we need this limit to be less than 1 in order for the series to converge, we set the absolute value of the limit to be less than 1:
|-(n+7)/(n+8) * x| < 1
Simplifying further, we have:
(n+7)/(n+8) * |x| < 1
Now we consider different cases:
Case 1: x = 0
If x = 0, then the series simplifies to ∑ 0, which is a convergent series.
Case 2: x ≠ 0
If x ≠ 0, we can divide both sides of the inequality by |x|:
(n+7)/(n+8) < 1/|x|
Now, we solve for the limit:
lim(n→∞) (n+7)/(n+8) = 1
So, 1/|x| < 1, which gives us:
|x| > 1
This means that the series converges when |x| > 1.
Therefore, the interval of convergence is (-∞, -1) ∪ (1, ∞).
Finally, the radius of convergence, R, is the distance from the center of the interval of convergence, which is 0 in this case, to either edge. So, R = 1.
Hence, the radius of convergence is 1 and the interval of convergence is (-∞, -1) ∪ (1, ∞).