To determine the radius of convergence and the interval of convergence for the given series, we can use the ratio test. Let's analyze the series:
∑ (-1)^n (x-2)^√n
Applying the ratio test:
lim┬(n→∞)|((-1)^(n+1) (x-2)^√(n+1)) / ((-1)^n (x-2)^√n)|
= lim┬(n→∞)|(x-2)^(√(n+1)-√n)|
= |x - 2|·lim┬(n→∞)|(n+1)^√n / n^√(n+1)|
Taking the limit:
lim┬(n→∞)|(n+1)^√n / n^√(n+1)| = 1
Therefore, the ratio test gives a value of 1, which does not provide any information about the convergence or divergence of the series. In such cases, we need to use additional methods to determine the convergence properties.
Let's consider the series when x = 2. In this case, the series simplifies to:
∑ (-1)^n (2-2)^√n
∑ 0
Since all terms of the series are zero, it converges for x = 2.
Next, let's consider the series when x ≠ 2. For the series to converge, the terms must approach zero as n goes to infinity. However, since the series contains (-1)^n, the terms do not approach zero, and the series diverges for x ≠ 2.
Therefore, the radius of convergence R is 0 (since the series converges only at x = 2), and the interval of convergence is {2}.
As for the absolute and conditional convergence, since the series diverges for x ≠ 2, it does not converge absolutely or conditionally for any interval other than {2}.