# Using the limit comparison test

Given the infinite series: $$\sum^{\infty}_{n=1}\frac{1}{2n+3}$$ Determine whether this series converges.

The answer key used the integral test to determine that no, this series does not converge.

I came at this problem differently. I first tried using the comparison test with $$\frac1n$$ which was inconclusive. I then tried the limit comparison test - again with $$\frac1n$$. I got a limit of $$\frac12$$. Because this is a finite, positive number - the limit diverges.

As a beginner, I am simply unsure that my method was legitimate - after all - its a fifty fifty chance of getting it right:) So, I am asking here- did I find the answer using a legitimate method?

$$\frac{1}{2n + 3} \gt \frac{1}{2n + 4} = \left(\frac{1}{2}\right)\left(\frac{1}{n + 2}\right) \tag{1}\label{eq1A}$$
and $$\sum_{n=1}^{\infty}\frac{1}{n + 2}$$ diverges.