I have to find if the following series converge:
a) Sum from k=1 to k=inf of k^k*e^(-k^2)
b) Sum from k=1 to k=inf of (1/sqrt(k))*e^(-2*sqrt(k))
For a I told that since k^k*e^(-k^2) is not equal to zero for k>=1 we can use the ratio test. I let a_k=k^k*e^(-k^2) then a_(k+1)/a_k=(((k+1)/k)^k)*(k+1) * e^(-2k+1) after some calculations. Then lim as k tends to inf of a_(k+1)/a_k is equal to 0<1 so our series converges absolutely by the ratio test.
Can anyone confirm my work and give me some guidance for which test should I use for part b? Thanks in advance!! Appreciate it!
c) I also have to determine the radius of convergence of the power series G(x)=Sum from k=1 to inf of (2k choose k)*x^k
For this I used the ratio test and found that abs(a_(k+1)/a_k)= 4 * abs(x) * (k+1/2)/(k+1) and as k tends to infinity then abs(a_(k+1)/a_k) tends to 4*abs(x)
Thus G(x) diverges for abs(x)>1/4 and converges absolutely for abs(x) < 1/4 so radius of convergence = 1/4
So basically if someone can check part a and c if I am correct and to guide me for part b.
Thanks in advance for any help!