# best convergence tests of polynomial series?

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

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

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

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

what is the test for others also?

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For the first, yes, the divergence test is the test to apply. For the second series, apply the root test, and for the third, compare it with $1/n\sqrt{n}$ and check convergence using $p$-series. –  Clayton Jan 4 '13 at 2:05

First $a_n\to ?$

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

Third does $\int a_n$ converge?

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actually I didn't understand your answer. –  Mustafa Jan 4 '13 at 2:12
@Okan In the first case I ask you to find where the $n$-th term of the series tends. Knowing this answer you will answer the question of the series convergence. Others are similar. –  Artem Jan 4 '13 at 2:15
For the third one, you can't use integral test since it's not monotonic. Anyway, you can compare it to $\sum \frac{1}{n^{3/2}}$. –  Amihai Zivan Jan 4 '13 at 7:13

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 $(2/5)^n$.

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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.

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