in the following series x is a real number. In each case use the ratio test to determine the radius of convergence of the series. Analyze the behavior of the series at the endpoints in order to determine the interval of convergence.
A) (nx^n)/(n^2 + 2)
B)((n^2)(x-2)^n)/2^n
C) ((4^n)(x^n))/(n^2)
To determine the radius of convergence using the ratio test, we need to evaluate the limit of the absolute value of the ratio of consecutive terms. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is exactly 1, the ratio test is inconclusive and we need to use additional tests.
Let's apply the ratio test and analyze the behavior of each series at the endpoints:
A) For the series A: (nx^n)/(n^2 + 2)
The ratio of consecutive terms is given by:
R = |((n+1)x^(n+1))/(n^2 + 2)| * |(n^2 + 2)/(nx^n)|
= |(n+1)/(n)| * |x|
= (n+1)/n * |x|
Taking the limit as n approaches infinity, we get:
lim (n+1)/n * |x| = |x|,
We need this limit to be less than 1 for convergence. Therefore, we have |x| < 1. This means the series A converges on the interval (-1, 1).
B) For the series B: ((n^2)(x-2)^n)/2^n
The ratio of consecutive terms is given by:
R = |((n+1)^2)(x-2)^(n+1)|/2^(n+1) * |2^n/(n^2)(x-2)^n|
= |(n+1)^2 * (x-2) / 2| * |n^2/(2^n)|
= (n+1)^2/2 * (x-2) * n^2/2^n
Taking the limit as n approaches infinity, we get:
lim (n+1)^2/2 * (x-2) * n^2/2^n = 0.
Since the limit is 0, the series B converges for all real numbers x. The radius of convergence is infinite.
C) For the series C: ((4^n)(x^n))/(n^2)
The ratio of consecutive terms is given by:
R = |((4^(n+1))(x^(n+1)))/(n+1)^2| * |(n^2)/((4^n)(x^n))|
= |4(x)(n^2)/((n+1)^2)|.
Taking the limit as n approaches infinity, we get:
lim 4(x)(n^2)/((n+1)^2) = 4x.
We need this limit to be less than 1 for convergence. Therefore, we have |4x| < 1. This means the series C converges on the interval (-1/4, 1/4).
So, the radius of convergence for series B is infinite, and the intervals of convergence for series A and C are (-1, 1) and (-1/4, 1/4) respectively.