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#.
