In the Comparison Tests section of my textbook, I am tasked with determining the convergence of the series $$\sum^{\infty}_{n=1}\frac{2}{\sqrt{n}+2}$$
I will argue the series diverges using the "Limit" Comparison test.
Consider $$a_n=\frac{2}{\sqrt{n}+2}$$ and $$b_n=\frac{1}{\sqrt{n}}$$ Note: The latter series $b_n$ is a divergent $p$-series with $p=\frac{1}{2}.
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If $$\lim_{n\rightarrow\infty}\frac{a_n}{b_n}=c$$ where $c>0$, then either both $a_n$ and $b_n$ converge, or both $a_n$ and $b_n$ diverge.
Thus $$\lim_{n\rightarrow\infty}\frac{a_n}{b_n}=\lim_{n\rightarrow\infty}\frac{2\sqrt{n}}{\sqrt{n}+2}=\lim_{n\rightarrow\infty}\frac{2}{1+\frac{2}{\sqrt{n}}}=2$$
Since the above limit existed as a finite value $c=2>0$ and since $\sum b_n$ diverges ($p$-series with $p=\frac{1}{2}$):
$$\sum^{\infty}_{n=1}\frac{2}{\sqrt{n}+2}\space\text{diverges}$$
However, from the beginning I can see the asymptotic equivalence: $$\frac{2}{\sqrt{n}+2}\sim\frac{1}{\sqrt{n}}$$ Why do I need the limit comparison test when I can deduce the result quickly upon inspection? Thanks in advance!
Answer
Note that $$\frac{2}{\sqrt{n}+2}\geq \frac{1}{\sqrt{n}}$$
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