determine convergence or divergence for the given series

Question

determine convergence or divergence for the given series. LIST which tests were used for each problem.

1. ∑k²e^-k for k = 1 to ∞
2. ∑n²e^{-n^3} for n = 1 to ∞
3. ∑(1/(n^3) + 1/3^n) for n = 1 to ∞
4. ∑(1/(ksqrt(k²+1))) for k = 1 to ∞
5. ∑(3ⁿn²)/n! for n = 1 to ∞
6. ∑(2^k-13^k+1)/k^k for k = 1 to ∞
7. ∑(n²+1)/(n³+1) for n = 1 to ∞

Answer 1

I will answer #6. Then you need to work some of these on your own and ask a question about those on which you are stumped. If you don't know how to do any of them, you need to go back and study and/or get formal tutoring help. It will do you NO GOOD for me to answer all of them for you.

In #6 the summand should be written 4*6^k-1/k^k.

At some point the fraction becomes less than one

The ratio test will show that this series converges.

For number 4, the comparison test works. Each term is less than 1/k² which converges; therefore the series converges.