Question

In Exercises \(3-6,\) find the radius of convergence of the power series. $$ \sum_{n=1}^{\infty} \frac{(2 x)^{n}}{n^{2}} $$

Short answer

The radius of convergence of the power series is \(\frac{1}{2}\).

Step-by-step solution

Apply Ratio Test

To evaluate the radius of convergence, the ratio test can be employed. According to the ratio test, calculate the following: \[ \lim_{n \to \infty} \left| \frac{\frac{(2x)^{n+1}}{(n+1)^2}}{\frac{(2x)^n}{n^2}} \right|\] Simplify this to: \[ \lim_{n \to \infty} \left| \frac{2x(n^2)}{(n+1)^2} \right|\]

Calculate the Limit

In the next step, compute the limit. As \(n \to \infty\), we can assume \(n=n+1\) in the limit, effectively cancelling them out. This gives us: \[ \left| 2x \right|\]

Use Ratio Test Result

For absolute/convergent series, this limit (also known as the common ratio) should be less than 1, according to the ratio test. Therefore: \[ \left| 2x \right| < 1\] Dividing by 2 gives the final result: