calculus help plz & thanks?

Question

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

1) summation notation symbol (with n=1 on bottom and infinity on top) of x^n/sqrt of n

2) summation notation symbol (with n=0 on bottom and infinity on top) of x^n/n!

3) summation notation symbol (with n=2 on bottom and infinity on top) of ((-1)^n) x (x^n/4^n ln(n))

4) summation notation symbol (with n=1 on bottom and infinity on top) of n!(2x-1)^n

show steps/work plz..thanks

Answer 1

take abs val an apply ratio test
a_n+1/a_n = IxI *sqrt((n+1)/n)) ==> IxI<1 so r=1
2) IxI *1/(n+1)==>0 so r = infinity ( the series represents e^x)
3) IxI/4 so IxI <4 r=4
4) (n+1)I2x-1I ==> infinity so r=0 never converges