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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?

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2 Answers 2

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Yes, you used the limit comparison test correctly

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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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