Need help with power series (radius of convergence)

Infinite series (n=1) of ((3^n)(x+4)^n)/√n
The radius of convergence is 1/3 but I keep getting 3. Can anyone help me?

By the Ratio Test,
r = lim(n→∞) |[(3^(n+1) (x+4)^(n+1))/√(n+1)] / [(3^n (x+4)^n)/√n]|
..= 3|x + 4| * lim(n→∞) √n / √(n+1)
..= 3|x + 4| * lim(n→∞) 1/√((n+1)/n)
..= 3|x + 4| * lim(n→∞) 1/√(1 + 1/n)
..= 3|x + 4|.

So, this series converges when |r| = 3|x + 4| < 1
==> |x + 4| < 1/3; thus the radius of convergence is 1/3.

I hope this helps!

Use the Root Test
{[(3^n)*(x+4)^n]/sqrt(n)}^(1/n)
{(3^(n/n)*(x+4)^(n/n))/sqrt(n)^(1/n)}
3|x+4|/{n^(1/2n)}
I need to find lim{n-->infinity} (n^(1/2n))
Let L=n^(1/2n)
ln(L)=ln(n)/(2n)
Use L'hopital's rule on ln(n)/2n
{d/dn(ln(n))}/d/dn(2n)
1/n/(2)
1/2n-->0 as n-->infinity
Therefore L=e^0=1
So the limit of the "nth root" is
3|x+4|
For this to converge, recall that the root test requires it to be <1 (less than 1)
3|x+4|<1
|x+4|<1/3
From this inequality, we see that r=1/3