Answers:
1. Yes. By using the ratio test one finds 2^(n+1)*n!/(2^n(n+1)!) which becomes 2/(n+1). By taking the limit as n goes to infinity we find it equals 0, which is less than 1, so it converges.
2. No, the first term is 16^-1, then 16^0, then 16^1, then 16^2-1, then 16^3, then 16^4. The fourth term does not follow the pattern of a ratio of 16, so it is not geometric.
3 & 4. P-series. Is can be rewritten as (1/6)(1/n^3) since 1/6 always converges and 1/n^3 converges by p-series, it will converge.
5 & 6. The nth term divergence test is most appropriate, since the limit of 4n as n goes to infinity is infinity, therefore it diverges.
7. The radius of convergence of cos(xn) is infinty since it converges no matter what value is multiplied by n.
8. (-1/2, 1/2). The formula 1/(1+r) is the sum of a geometric series and converges when |r|<1, so |2x| <1 finally, -1/2<x<1/2.
9. The radius of convergence is 1/2 because (1/2 - (-1/2))/2 = 1/2.
10. No, there are no values where the function converges conditionally.
2. No, the first term is 16^-1, then 16^0, then 16^1, then 16^2-1, then 16^3, then 16^4. The fourth term does not follow the pattern of a ratio of 16, so it is not geometric.
3 & 4. P-series. Is can be rewritten as (1/6)(1/n^3) since 1/6 always converges and 1/n^3 converges by p-series, it will converge.
5 & 6. The nth term divergence test is most appropriate, since the limit of 4n as n goes to infinity is infinity, therefore it diverges.
7. The radius of convergence of cos(xn) is infinty since it converges no matter what value is multiplied by n.
8. (-1/2, 1/2). The formula 1/(1+r) is the sum of a geometric series and converges when |r|<1, so |2x| <1 finally, -1/2<x<1/2.
9. The radius of convergence is 1/2 because (1/2 - (-1/2))/2 = 1/2.
10. No, there are no values where the function converges conditionally.