determine whether the following series converges or diverges; state which test you are using:

a)(k^2*2^K)/(k+1)!

b)3^(k+1)/[(k+1)^2*e^k)]

c)ln(k)/k

d)(2k+1)/sqrt(k^5+3k^3+4)

Question

determine whether the following series converges or diverges; state which test you are using:

a)(k^2*2^K)/(k+1)!

b)3^(k+1)/[(k+1)^2*e^k)]

c)ln(k)/k

d)(2k+1)/sqrt(k^5+3k^3+4)

anonymous · Answer

For the first one use the ratio test by taking the ratio of:
$$\lim_{k \rightarrow \infty} \left| \frac{a_{k+1}}{a_k} \right|<1$$
means convergence.
And I get the limit goes to zero.
For the THIRD one use the integral test. It is a decreasing, positive continuous function over k. So integrate it from k=1 to infinity, if it converges then the sum converges. For the LAST ONE, use the limit comparison test, same with the second I believe.