Answer
Diverges
Work Step by Step
The given series can be summed up as: $\dfrac{1}{200}+\dfrac{1}{210}+\dfrac{1}{220}+.....=\Sigma_{n=1}^{\infty} \dfrac{1}{10 n+200}$
Let us consider that $a_n=\Sigma_{n=1}^{\infty} \dfrac{1}{10 n+200}$ and $b_n=\Sigma_{n=1}^{\infty} \dfrac{1}{n}$
We can see that $a_n \leq b_n$ and $b_n$ shows a p-series with $p=1$. This implies that the series $b_n$ diverges by the p-series test.
Next, we have $\lim\limits_{n \to \infty} \dfrac{a_n}{b_n} =\lim\limits_{n \to \infty} (\dfrac{1}{10n+200}) \times n$
or, $=\lim\limits_{n \to \infty} \dfrac{n}{10 n+200}$
or, $=\lim\limits_{n \to \infty} \dfrac{1}{10+200/n}$
or, $=\dfrac{1}{10}$
Hence, we can see that the given series diverges by the limit comparison test.
