The problem you've provided seems to involve finding the radius and interval of convergence for a power series. Here's the series you've provided:

$\sum_{n=1}^{\infty} \frac{n(x-5)^n}{(2n-1)5^n}$

To find the radius and interval of convergence, we can use the ratio test to determine the radius of convergence. Let's go through the process step by step.

1. Ratio Test: The ratio test involves finding the limit of the ratio of consecutive terms. For the series $\sum a_n$, the ratio test tells us:

   $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

   If $L < 1$, the series converges. If $L > 1$, the series diverges, and if $L = 1$, the test is inconclusive. For power series, this test can help find the radius of convergence $R$ as $R = \frac{1}{L}$.

2. Applying the Ratio Test: For the given series, $a_n = \frac{n(x-5)^n}{(2n-1)5^n}$. We need