1. The problem is to find the radius and interval of convergence of the series $$\sum_{n=0}^\infty (x+5)^n$$ and determine for which values of $x$ the series converges. 2. This is a geometric series with common ratio $r = x+5$. 3. A geometric series converges if and only if $$|r| < 1$$. 4. Applying this to our series, we have $$|x+5| < 1$$. 5. Solve the inequality: $$-1 < x+5 < 1$$ 6. Subtract 5 from all parts: $$-6 < x < -4$$ 7. The radius of convergence $R$ is the distance from the center $-5$ to either endpoint, so: $$R = 1$$ 8. The interval of convergence is $$(-6, -4)$$. 9. Check endpoints: - At $x = -6$, the series becomes $$\sum_{n=0}^\infty (-1)^n$$ which does not converge. - At $x = -4$, the series becomes $$\sum_{n=0}^\infty 1^n$$ which diverges. 10. Therefore, the series converges for $$x$$ in the open interval $$(-6, -4)$$ only. Final answer: Radius of convergence is $1$, interval of convergence is $$(-6, -4)$$, and the series converges for $$x$$ such that $$-6 < x < -4$$.