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?


2 Answers 2


Yes, you used the limit comparison test correctly


As the answer states, you used the limit comparison test correctly. Note that another way you could have used the comparison test is with

$$\frac{1}{2n + 3} \gt \frac{1}{2n + 4} = \left(\frac{1}{2}\right)\left(\frac{1}{n + 2}\right) \tag{1}\label{eq1A}$$

and $\sum_{n=1}^{\infty}\frac{1}{n + 2}$ diverges.


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