Answer
Convergent
Work Step by Step
The given series can be summed up as: $\dfrac{1}{201}+\dfrac{1}{208}+\dfrac{1}{227}+.....=\Sigma_{n=1}^{\infty} \dfrac{1}{n^3+200}$
We will apply the limit comparison test with $a_n=\Sigma_{n=1}^{\infty} \dfrac{1}{n^3+200}$ and $b_n=\Sigma_{n=1}^{\infty} \dfrac{1}{n^3}$
We can see that the series $b_n$ shows a convergent p-series with $p=3 \gt 1$.
Next, we have $\lim\limits_{n \to \infty} \dfrac{a_n}{b_n} =\lim\limits_{n \to \infty} [\dfrac{ \dfrac{1}{n^3+200}}{1/n^3}]$
or, $=\lim\limits_{n \to \infty} \dfrac{1}{1+200/n^3}$
or, $=1$; a finite and positive term
Hence, we can see that the given series is convergent by the limit comparison test.
