best convergence tests of polynomial series?

Question

$$ \sum_{i=1}^\infty {2n+3\over 5n+1} $$

$$ \sum_{i=1}^\infty \left({2n+3\over 5n+1}\right)^n $$

$$ \sum_{i=1}^\infty {\sin^2n\over n \sqrt{n}} $$

for 1st one is it okay to apply divergence test?

what is the test for others also?

thanks in advance

Answer 1

First $a_n\to ?$

Second $(a_n)^{1/n}\to ?$

Third does $\int a_n$ converge?

Answer 2

Notice that if the terms don't go to zero, the series can't converge. For the third one, there's a simple comparision test to a series I expect you know about. For the second, compare to the sum of $(2/5)^n$.

Answer 3

About the thrid one, you can also use this fact that $$\lim_{n\to\infty}n^{\frac{-1}{2}}\frac{\sin^2(n)}{n\sqrt{n}}=1\neq$$ so it diverges. For others @Artem's answer is practical.