Question

Let \[ a_n = \frac{(1 + (-1)^{n})}{2n} + \frac{(1 + (-1)^{n-1})}{3n}. \] Then the radius of convergence of the power series \[ \sum_{n=1}^{\infty} a_n x^n \text{ about } x = 0 \text{ is ..............} \]

Hint:

The ratio test is often used to determine the radius of convergence for power series.

Answer 1

Correct Answer: 2

Step 1: Apply the ratio test.
To determine the radius of convergence, we apply the ratio test. The ratio test involves computing the limit: \[ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|. \] The radius of convergence \( R \) is then given by: \[ R = \frac{1}{L}. \]

Step 2: Simplify the terms.
We simplify the given expression for \( a_n \) and calculate the ratio test. After simplifying, we find that the radius of convergence is \( R = 1 \).

Step 3: Conclusion.
The radius of convergence is \( \boxed{1} \).