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I am given the following comparison test question on my homework. Based on just plugging in the numbers, the first condition in all three answers is wrong, since it should say the original function is less than or equal to the comparison function, not just less than. The first two terms, for n=2 and n=3, are equal to 1/2.

Am I missing something?

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You are right, it should say $\frac 1{n!}\color{blue}{\leq} \frac 1{n(n-1)}$ rather than $\frac{1}{n!}\color{red}{<} \frac 1{n(n-1)}$, otherwise all the "since" statements will be wrong for $n=2,3$. After the change, I think you will see that only one option is correct.

While it is good to be picky over these things, I think that the question is set up to understand when the comparison test works, so the given example in the question is not as important as the "principle of comparison" itself. If you have found the answer after correcting the question, I think you have understood the comparison principle well.

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