To find the radius of convergence we can apply the ratio test, stating that a necessary condition for a series sum_(n=1)^oo a_n to converge is that:
L = lim_(n->oo) abs (a_(n+1)/a_n) <= 1
If L < 1 the condition is also sufficient and the series converges absolutely.
Let's calculate the ratio:
abs (a_(n+1)/a_n) = abs ( frac (x^(n+1)/5^sqrt(n+1)) (x^n/5^sqrt(n))) = abs(x) 5^sqrt(n)/5^(sqrt(n+1)) = abs(x) 5^((sqrt(n) - sqrt(n+1)))
Now we have:
sqrt(n) - sqrt(n+1) = (( sqrt(n) - sqrt(n+1)) ( sqrt(n) + sqrt(n+1)))/ (sqrt(n) + sqrt(n+1)) = (n-(n+1))/(sqrt(n) + sqrt(n+1)) = -1/(sqrt(n) + sqrt(n+1))
and as a result:
lim_(n->oo) (sqrt(n) - sqrt(n+1)) = lim_(n->oo) -1/(sqrt(n) + sqrt(n+1)) = 0
lim_(n->oo) 5^(sqrt(n) - sqrt(n+1)) = 1
so that:
lim_(n->oo) abs (a_(n+1)/a_n) = abs(x)
which means that the series is absolutely convergent for abs(x) < 1 and divergent for abs(x) >1.
