Find the radius of convergence and interval of convergence of the series.

infinity(sigma)n=1

(x^(n+4))/(sqrt(n))

R=_______


I=[ ______ , _______ )

By applying the Ratio Test, we see that:
lim (n-->infinity) (a_n+1)/(a_n)
= lim (n-->infinity) [x^(n + 5)/√(n + 1)]/[x^(n + 4)/√n], since a_n = x^(n + 4)/√n
= x * lim (n-->infinity) √n/√(n + 1)
= (x)(1)
= x.

So, the series converges absolutely for:
|x| < 1 ==> -1 < x < 1.

|x| < 1 implies that R = 1.

Then, by checking the endpoints:
(a) x = -1 ==> sum(n=1 to infinity) (-1)^n/√n, which converges by the AST
(b) x = 1 ==> sum(n=1 to infinity) 1/√n, which is a divergent p-series.

Therefore, the series converges for x E [-1, 1).

I hope this helps!